Going back to the technology, the simple claim of brainwave entrainment is that, by listening to these audios, usually with stereo headphones and eyes closed, your brain will be naturally and effortlessly led into a specific brainwave state as designated by the maker of the audio (ie: it could be alpha, theta, delta, gamma, etc., depending on what the audio is designed to do.)  The other most basic claim is that when listening to these audios, the right and left hemispheres of the brain synchronize and you experience whole-brain functioning.
When the brain is stimulated with pulsed sounds (neuro-electrical activity via the nerves originating from the ears), the overall activity of the brain will respond to and align with these pulses. By selecting the desired rate, the brain — via the frequency following response (entrainment) — can be naturally induced towards the selected brainwave state.
But here is where I’d like to chime in my own two cents.  As a user of brainwave entrainment products for over 8 years now, you would obviously have to guess that I think this technology is fabulous.  Even my earlier experiences with brainwave entrainment technology using less refined binaural beats were nothing short of miraculous for me.  It’s a very fascinating experience to go from not being able to meditate at all to being able to sit with your eyes closed and headphones on and effortlessly slip down into deep brainwave states that were previously only reserved for years-long experienced meditators.
Much of what is currently written about brainwave entrainment, especially on the internet, is simply … incorrect.  It seems par for the course that when something of huge value is discovered as a potential ‘internet product’ that it is going to be marketed within an inch of it’s life. And often with less than stellar regard for actual science, research, or factual knowledge. 
Deep relaxation is another major benefit resulting from the brains cortical frequency following response while using brainwave entrainment. As the brain tends to mimic the rate of frequency it is exposed to, thus allowing it to enter into that brainwave state, this helps explain why this technology can produce benefits commonly found with meditation.
Entrainment is a term originally derived from complex systems theory, and denotes the way that two or more independent, autonomous oscillators with differing rhythms or frequencies, when situated in a context and at a proximity where they can interact for long enough, influence each other mutually, to a degree dependent on coupling force, such that they adjust until both oscillate with the same frequency. Examples include the mechanical entrainment or cyclic synchronization of two electric clothes dryers placed in close proximity, and the biological entrainment evident in the synchronized illumination of fireflies.[13]
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