This is a brand of what's called 'greasing the groove,' a common way of overcoming limitations in body weight movements, more effectively applied to pull-ups. (For reference, pull ups are palms away, and much more difficult than chin ups where your palms are toward you).

Essentially what you do is you try to do a set of as many as you can, stopping just before failure--meaning your final rep should be a complete rep, not a forced kicking and screaming half rep. Over the course the day you continually do more sets, by the evening you'll probably have more than a hundred completed. In doing so, you increase your total number gradually and your body adapts.

It works well for pull ups, I used a method at the gym where I shoot for a total number, such as 30, and do as many sets as necessary to get to that number. I've found it to be effective, I consider myself bad at pull-ups and pull downs, but it has taken my reps from five consecutive to about fifteen consecutive in a little under two months.

However, you must consider this doesn't have a linear translation to push-ups. Pull-ups you are using 100% of your body weight, the goal is to eventually do repetitions while adding weight to your initial bodyweight. In a push-up you're only pushing about 40-60% of your weight, so after about twelve consecutive repetitions it's merely inducing fatigue. Twelve reps is a feat that should be within reach even by a relative novice after a few weeks, at most, months. Twelve pull-ups, on the other hand, can take quite a lot of time for someone to hit.

Furthermore, additional weight to a pull-up is more easily achieved than that of a push-up, it's nearly impossible to place weight safely on your back between your shoulder blades. By doing so, you're simply turning it into a bench press.

Some people like push-ups for burnout effect, where they do a heavy set on bench or dumbbells and then drop right into wide-gripped push-ups to activate pectoral muscles. Rarely does anyone use them to great result for muscle growth. In my opinion, clappers (plyometrics) will produce better results and have a more profound affect on other conditioning exercises.

In a nutshell, if you have fundamental problems with push-ups--such as an inability to do them, or improperly doing them (nose touches floor, ass down)--you should consider this, but if you are capable of push-ups you should look towards weighted chest movements.