The way I describe it to my students is using the idea of popping someone with a towel. If you don't pull back slightly at the end, you get no pop, pull back slightly as the towel's tip is almost extended and you accelerate the tip of the towel.
This is the same for the punch, this slight pull back while the fist is still going forward gives tremendous acceleration and focuses the energy in a tighter pattern.
I've always had a problem with likening the wave/snap motion to a towel or whip. It just doesn't make sense to me, though I don't have the physics knowledge to outright disprove it.
Basically, the human arm or leg simply isn't as flexible as a towel or whip, so I don't see how a human limb can transfer energy in the same way
. Furthermore, the power of a towel/whip comes from the "flicking" motion. But you don't really "flick" punches and kicks, you drive them straight. You can "flick" a backhand, and some kicks, but "normal" punches and kicks are "straight" (at least, the ones I used to practice in karate were).
Also, the way I understand whips (and to a lessor extent towels), the reason the wave accelerates is because the whip tapers from handle to tip. Energy is conserved, thus the motion accelerates as the mass of the whip decreases. The human body isn't built this way. Furthermore, a whip can get moving really, really fast. The "crack" of the whip comes from the tip moving faster than the speed of sound, a mini-sonic boom. The human body certainly can't do that.
I'm sure there are other "wave"-like actions the human body can make, but I don't see how they can be the same as a towel or whip.