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I've come across software that claims to improve one's sleep and to help falling asleep in the first place, by listening to an audio track with spoken words, music, and ambient sounds. Here's one example of an iPhone app. The vendor also offers a reverse product with a power-nap effect.
There are lots of reviews that claim this stuff works, and maybe it does, but maybe the reviewer was tired anyway or it's a placebo effect.
My attempts to find any evidence for or against the use of audio in general, or binaural beats in particular, as a treatment for insomnia revealed no strong evidence either way (making this answer rather less than definitive.)
However, this paper argues against the practice:
One should not watch television, listen to music or audio tapes, check e-mail, answer cell phones, solve family problems, or read in bed.
This is justified on the basis of "sleep hygiene" - the idea of training the mind to associate lying in bed with sleeping.
That said, I speculate that for some insomniacs, having a set routine which involves lying down, shutting your eyes and ignoring some patternless noise from a phone, may actually serve as a higher-quality routine for sleep-hygiene that using the same phone to watch videos and answer questions on StackExchange web sites while lying in bed.
Two other studies were of interest:
This meta-study grouped together a pool of unrelated self-help techniques for insomnia, and found their overall strength was poor-to-moderate. Their conclusion:
Self-help fits well as a first step in a stepped care model for insomnia.
I read this as suggesting running to the doctor need not be your first remedy, although your doctor has more effective tools available if you need them.
I'll throw in another study from 1997 even though I haven't found the original article, the abstract doesn't contain enough relevant information and overall, it looks rather suspect. It claims that playing music previously generated from the data in a patient's own EEG can improve insomnia. Definitely something I would like to see independently replicated and more details of the experimental method before I could accept it.
This argument presumes that you cannot sleep on your own in a relaxed state and need assistance to fall asleep.
In order for software to cause sleep it would have to inititate the stages of sleep in a fashion beyond your control. By cause I mean 'to render you unconscious', as per the Hollywood image of someone being chloroformed, or the very real use of anesthesia at the doctor's office. Putting an individual to sleep in this manner should be as automatic as a drug, but still step by step, as the process should naturally occur. I will presume here that no other method of establishing sleep is being used (especially pharmaceutical at this point).
The software should, if successful in initiating sleep, cause your brain to produce High Amplitude Theta Waves. According to about.com's article on the stages of sleep, this is stage 1. If this does not happen because of the software alone, then it did not cause Stage 1 Sleep. "Initiating" in this sense is a forceful start of the Sleep cycle that allows the body to take over after it "gets the ball rolling". There are also no claims or explanations about how the product protects the Sleep Cycle from being interrupted.
There is a comment on the page linked in the product review provided where a commenter claims to be a light sleeper and cannot fall asleep with this Software active. This is clearly an instance of the Software being unable to fulfill what it was deisgned for. I can definitely say, this product does not work for everyone, but may remove stress in some or most cases, if the comment is accurate. The Wired review article also linked in your question states something very similar.
Sara Mednick, a sleep researcher at the Salk Institute and the author of Take a Nap, Change Your Life, says there's no hard science proving that pzizz and programs like it actually improve sleep. But the promise of a planned, computer-controlled, 30-minute nap to refresh the system is alluring to the modern psyche.
Verdict: It's all in your head. It may trick you into sleeping, but you're really falling asleep on your own. You could possibly achieve the same sleeping success by listening to an extremely boring lecture, or sitting on a comfy couch after eating a large meal, but there's nothing in the product listed that would cause you to sleep directly. This product only assists in relaxing the listenener. This software is more accurately a "Stress Removal Aid" and not a "Sleep Aid". If you fall asleep to the product linked above, there's a good chance that you could have fallen asleep on your own.
Inspiring words, enchanting music, delightful sound effects and binaural beat. There is a question about binaural beats.