The claw essentially allows you to press the c-stick with the same speed that doing it with the control stick allows. The principle behind it is that for some chars - especially falcon w/uair, (mario too, but.. just no) u cannot hit a standing enemy with a FH uair (without insanely fast hands) because it takes too long for your thumb to get to the c-stick. Some (like me) resort to using the control stick, but there is a risk of doing ur 2nd jump during the uair, which can lead to death/droppage of combo. The solution, stick your 2nd finger on the jump button, thumb on the c-stick.
If your WDing is hype, the claw makes it alot more difficult, but not impossible. You will have to learn how to SH again, lol, but its not as hard this time. Mainly, it just means that you can press the aerial button of your choice alot faster than if you were to move your thumb from jump -> cstick. I would like to learn the claw, but i substituted in using the control stick instead. Many people use the c-stick with marios uair blinds, its so much slower its kind of funy. If i could claw the blind it would be the same speed as me doing it with the control stick, and faster then people using the c-stick
for a rough comparison of the timing, Brown Mario vs Submaburu (?), or something like that. At the end, Brown Mario pulls off an uair blind, which looks super sexy, and super fast. Now look at eggz combo video, notice the difference of about 4/5 frames (maybe less i don't know) in their uairs. There are always going to be people saying "oh, yeah, no way i use the c-stick and its just as fast as the control stick," but for the majority that is not the case. Be accurate with the control stick or resort to the claw, its a personal choice. Imo the claw is safer, and may even allow you an extra frame of speed, maybe even 2.
lol i always do this when i write **** down, i just convinced myself into learning claw