Why does the garlic peeling method work when the chef is doing it? See detail.

Practice, Practice, Practice.
I just press down on the separate segments with the flat of a knife to slightly crush it.

I also like this trick here

I don’t know, but I am going to try it next time I buy a head of garlic. The trick seems to be cutting the root away from the cloves first.

There is a slight twist in two axes of your wrist, while preventing the knife from slipping from the clove. i.e. it takes practice

I tried it too after I posted the question it worked, kind of.