In 1956, the famous neuroscientist W. Gray Walter published the results of studying thousands of test subjects using photic stimulation, showing their change in mental and emotional states. He also learned that photic stimulation not only altered brainwaves, but that these changes were occurring in areas of the brain outside of vision. In Walter’s words:
Literally embrace whatever it is you're going through and then let it go. Try doing a tai chi exercise known as "embracing the tiger," where you take your arms, spread them wide, put your hands together and then draw them—and everything around you—toward your navel, the center of your being. Doing this allows you to take the good with the bad. Then reverse your hands and push them out, releasing your tension. When you can control stress, it can no longer control you. (Follow it with this progressive muscle relaxation technique to reduce stress.)

When tuning instruments that can produce sustained tones, beats can be readily recognized. Tuning two tones to a unison will present a peculiar effect: when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible. As the two tones get further apart, their beat frequency starts to approach the range of human pitch perception[1], the beating starts to sound like a note, and a combination tone is produced. This combination tone can also be referred to as a missing fundamental, as the beat frequency of any two tones is equivalent to the frequency of their implied fundamental frequency.

You may recognize the "binaural recording" from ASMR, where the technique is often used. This explains why, if you search for "binaural" on YouTube, you get a mixture of two entirely different kinds of results: binaural beats and binaural ASMR recordings. It gets especially confusing in that there's some questionable and/or unexplored science surrounding each one.

“One of the best ways to explain how brainwave entrainment works is by observing the effect of a tuning fork on a guitar string,” says Naik. “Hit a tuning fork tuned to a particular sound frequency, such as the note C, and hold it close to the strings of a guitar. You will notice that the C string on the guitar starts to vibrate because it has entrained on to the same frequency of the tuning fork.”
The mechanism for this is that when your eyes or ears are exposed to a particular frequency of pulses or beats, the thalamus first distributes this information to the entire brain, including the visual and cerebral cortex where neural activity begins to synchronize to the incoming frequency, producing hemispheric synchronization and a balance of brainwave activity across the brain.
^ Trost W. and Vuilleumier P., Rhythmic entrainment as a mechanism for emotion induction by music: a neurophysiological perspective. In The Emotional Power of Music: Multidisciplinary Perspectives on Musical Arousal, Expression, and Social Control, Cochrane T., Fantini B., and Scherer K. R., (Eds.), Oxford, UK: Oxford University Press; 2013, pp213–225.

So, in summary, binaural beats certainly do not work the way the sellers claim, but there's no reason to think they're any less effective than any other music track you might listen to that effects you in a way you like. If they make you sleepy (like they all do for me), use them to go to sleep. If they relax you or get you amped, use them for that. But don't expect them to be any more effective than regular music. If someone you know claims that they are, put them to the test, and bust the myth.
However, those with ASD are sometimes also diagnosed with ADHD. Although they are different conditions, some of the symptoms can be intertwined, which it appears may be the case with your grandson. So I would start by trying out my tracks for ADHD. You can try those out for free on my YouTube channel and I’ve put them together in a playlist here: The corresponding MP3 versions for those tracks can be found on my site here: I hope that helps.
Because the brain filters and interprets reality in a split-brained way, we tend to see things as separate and opposed, rather than as connected and part of the oneness spoken of by the great spiritual teachers (and, in the last few decades, by quantum mechanical physicists). Thus, at a deep level, the dual structure of our brain, in conjunction with brain lateralization, predisposes us to see and experience ourselves as separate from, and often in opposition to, the rest of the world—instead of experiencing the elegant interconnectedness between us and everything else. Our childhood associations and programming build on this inborn tendency by training us to seek this and avoid that, to move toward pleasure and away from pain, to do good and not bad, and so on. The greater the lateralization in the brain, the greater the feelings of separation—and the greater the feelings of separation, the greater the fear, stress, anxiety, and isolation.
The mantra technique as taught by Michael at Project Meditation is an excellent example of an outward technique. Since the vast majority of people have somewhat active minds, this mantra technique is a good bet for most of us. Even mantra techniques have some differences. Some require rigid concentration while others are more relaxed, such as Michael’s technique. There are some that use chanting which is helpful for a person whose mind is hyperactive.
SPEAKER: You know those people who always seem to be smiling? What's their secret? Turns out a lot of the smiles come from saying sayonara stress. Want in on that action? Well, here are three ways to stress less so you can smile more. First up, play your stress away/ why should kids have all the fun? Your boss might not let everyone out for recess, but you can find 15 minutes every day to do something you really enjoy. Go to a driving range. Play ping pong, cards, or board games. Just keep it friendly. Remember, winning isn't everything. Number 2. Give it up for pet power. Have you ever seen anyone looking stressed when playing with an adorable puppy? Neither have we. Interacting with animals has been proven to lower stress in almost everyone. Simply petting a dog or a cat lowers your blood pressure considerably and reduces levels of the stress-causing hormone cortisol. So pet your pooch. Don't have a pet? Borrow a friend's, or visit a local shelter for some much-needed cuddle time. And the number-one way to stress less? Laugh it off. How does laughter love thee? Let us count the ways. Laughter fills your body and lungs with oxygen. It makes your brain release Mr. And Mrs. Happy Hormones, the endorphins. It bolsters your immune system and helps you, well, be well. It also helps your brain release natural pain relievers, and may even stop painful muscle spasms. It's true. He who laughs best stresses less. So learn to play, adopt a stray, and laugh away. You'll feel great. And it will show.
The Neuro Programmer website, as is typical, is full of testimonials and vague references to research. I tried to track down the research they referenced, but could not find anything published or peer-reviewed by the authors named or on the subjects indicated. For example, they cite one study by Thomas Budzynski, but the only thing published by him that is listed on Pub Med is a small study on biofeedback published in 1969. The 1999 study they reference is not listed. Also – Budzynski has connections to another company – Theta Technologies – which sells similar technology.  (The exact nature of his connection is not clear, he was at least a consultant. He is also listed as the director of research for another company, Synchromed, LLC.)  This looks like just another in-house worthless study to support the marketing of a product.

According to a recent study published in the British journal Heart, slow or meditative music is a proven stress buster, so set your dial to a soothing station during your commute. And, if you're stuck in a traffic jam, sneak in this quick exercise: Grab your steering wheel and clench the muscles in your fingers, arms, shoulders and back. Do this until your muscles begin to tremble (about 45 seconds), then release. You'll produce a wave of relief in your upper neck and arms all the way down to your fingers. Just make sure your foot is on the brake when you let go of the wheel! (FYI: pink noise is the newest tool for reducing stress.)

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When signals of two different frequencies are presented, one to each ear, the brain detects phase differences between these signals. "Under natural circumstances a detected phase difference would provide directional information. The brain processes this anomalous information differently when these phase differences are heard with stereo headphones or speakers. A perceptual integration of the two signals takes place, producing the sensation of a third "beat" frequency. The difference between the signals waxes and wanes as the two different input frequencies mesh in and out of phase. As a result of these constantly increasing and decreasing differences, an amplitude-modulated standing wave -the binaural beat- is heard. The binaural beat is perceived as a fluctuating rhythm at the frequency of the difference between the two auditory inputs. Evidence suggests that the binaural beats are generated in the brainstem's superior olivary nucleus, the first site of contra-lateral integration in the auditory system (Oster, 1973). Studies also suggest that the frequency-following response originates from the inferior colliculus (Smith, Marsh, & Brown, 1975)" (Owens & Atwater, 1995). This activity is conducted to the cortex where it can be recorded by scalp electrodes.  

In addition to understanding how exercise can help manage stress and the types of exercise to recommend for stress management, it is important to understand common barriers that might affect exercise participation in high-stress clients. Lack of time is the most commonly reported exercise barrier for individuals in general. A lack of motivation, fatigue, poor sleep habits, and poor dietary habits are factors associated with stress that can negatively impact exercise compliance and adherence (4). Common exercise barriers and stress-related health problems should be taken into consideration when developing an exercise prescription for high-stress individuals.

The various mental states of the individual are thought to take place across a varied range of frequencies, or brainwaves. By encouraging the frequency following process, entrainment is able to create positive change in the brain, through matching carefully-selected frequencies of light and/or sound. The stimulus enables the individual to access a different state of consciousness, which can be useful for a number of benefits including relaxation, anxiety management, stress reduction and more.
While a practical understanding of brainwaves has been around for as long as people have been singing, chanting, and drumming, a scientific view of the electrical activity inside the human brain was not published until 1924 when German psychiatrist Hans Berger developed a machine for sensing and recording activity in the brain by attaching small electrical sensors to the scalp of his patients and recording the resulting electrical activity. Berger’s inventions and discoveries were built upon the earlier work of Richard Caton who published animal studies on brainwave oscillations in 1875.
Today we're going to put on our headphones, kick back in the beanbag, and get mellow to the soothing sounds of the latest digital drug: binaural beats. These computer generated sound files are said to massage your brain and produce all sorts of effects, everything from psychedelic experiences to behavior modification. Let's point our skeptical eye at the science of binaural beats, and especially at some of the claims made for them.
There’s an aesthetic element to the music and there are a lot of drone-based sounds and time-stretched sounds, deep, rich bass sounds—there’s a spectrum of sounds within there—and what’s more, there’s a load of 3D processing that goes on to ensure that the mind doesn’t become habituated to them. There’s also a constant, subtle sense, a slow and gradual sense, of the sounds shifting around. And, there are layers of binaural beats.
With regard to listening without headphones specifically, you might find it irritating if you are someone who suffers from motion sickness, depending on how the track is created. If you were listening to binaural beats, they really need headphones to work properly. If you are in a room and closer to one speaker you may not even hear binaural beats properly, and if sitting off centre in the room generally, that may have unsettled you if you have motion sickness as the tones try to form a beat in your head. If it was just a standard isochronic tones theta track that shouldn’t be irritating if you have motion sickness.

... Several studies have looked at the possible effects of binaural beats within the alpha range on cognitive abilities. A significant improvement in cognitive processing, as measured by the Stroop Effect exercise, was found by a BB stimulation of 10.2 Hz frequency (Cruceanu & Rotarescu, 2013).Carter and Russell (1993)exposed 8 to 12 year old boys with learning disabilities to 8-week long 10 and 18 Hz BB stimulation sessions, and they found an improvement in Raven's progressive matrices and in a subtest of auditory sequential memory (Carter & Russell, 1993).McMurray (2006)assessed the effect of 7 and 11 Hz BB on alpha brainwave activity, working memory, and attention in healthy elderly people, who are known for experiencing gradual decrease in physiological alpha activity. The 2 minutes exposure to BB resulted in an altered electrical activity in the brain. ...
A word to the purist here. This binaural beat generator offers ten carrier frequencies, but humans only have two ears! Therefore, carriers on this generator will produce amplitude-modulated beating patterns inside each ear canal. This is not necessarily a bad thing, as it increases the perception of the overall beating pattern, and helps our brain catch up with the binaural beat. This explains why this generator produces stronger beats than any other available. If you are looking for a pure binaural beat generator instead - one without any intra-aural interference but a weaker stimulus - try our Harmonic Binaural Beat Generator; its carriers have been set to distant frequencies, in order to suppress any amplitude modulation between carriers.
While binaural beats are the most common technique we use for producing brainwave entrainment music, we do use other brainwave entrainment techniques such as isochronic tones, monaural beats and amplitude modulation. If these techniques have been used then you will find them mentioned in the written description of the music productions that include them. If not mentioned, then you can assume that binaural beats are the only brainwave entrainment technology in use.
“A quite different phenomenon results when stereophonic earphones are used and the signals are applied separately to each ear. Under the right circumstances beats can be perceived, but they are of an entirely different character. They are called binaural beats. . . . Binaural beats require the combined action of both ears. They exist as a consequence of the interaction of perceptions within the brain.”

Our brainwave states are natural. Guiding the brain into those states intentionally by using entrainment is just another way of experiencing those same natural states. Nothing “unnatural” is produced by the brain as a result of using entrainment. The only thing that is different is that when using entrainment we often go into the experience with either general expectations, or a well defined laundry list of specific expectations that we associate with entrainment. We have no such list going throughout our normal day when we’re not using entrainment.
Everyone experiences stress, and not all stress is bad. Individuals who experience acute episodic or chronic stress are at increased risk for developing stress-related health problems. Research supports the idea that exercise can improve the way the body handles stress, and it can provide a time-out from stressors. Exercise programs meeting the current recommendations for health included within a stress management program can be effective in stress reduction. However, it is important to consider the client’s stressors and physical activity barriers, activities the client will enjoy, and the exercise setting. Prescribing exercise for clients seeking stress management is recommended, but fitness professionals should recognize that some clients will need additional assistance for managing stress and major life stressors.
While the physiological and psychological processes being uncovered by brain science in the study of brainwaves is sometimes complex and still being investigated, certain basic principles are well-established, easily understood, and helpful for achieving the most effectively use of brainwave entrainment when seeking relaxation, improved sleep, lowering of anxiety, or other goals. This important background information and scientific knowledge presented here includes: 
We can’t always control sources of stress in our lives, but we change how we react to them. The good news is this: The human body is actually designed to experience and handle stress, which is exactly why our bodies react to it so strongly. With some practice, we have the power to learn to use certain elements of stress to our advantage (for example, the fact that stress keeps us more alert and attentive), while better controlling other negative reactions (like digestion problems or giving in to cravings for unhealthy foods).

It sounds like your brain might have initial trouble adapting to the binaural beats. The same thing happened to me when I listen to the Beta frequency (I got a headache), but then I discovered that I had the volume turned up too loud. I would say experiment with these frequencies, but at a low volume. See how you go. And if you continue to have anxiety, by all means, stop listening. Also, not all binaural beats are created equally. Some are amateurish, but the ones from Binaural Beats Meditation (mentioned and linked to in this article), are professionally created and the ones that I personally listen to. I love them (especially the Theta/Delta ones)! I hope this helps. :)
In 1984 medical researcher Dr. Gene W. Brockopp published a paper making several conclusions of audio and visual entrainment (AVE). Such conclusions were that hemispheric synchronization caused by AVE is related to increased intellectual functioning, practiced use of AVE overtime leads to a cumulative effect, and AVE may result in the recovery of early childhood experiences.