Singers wiggling their jaw

Why do some singers wiggle their jaw to do riffs/ and how do you do it 

I suppose they're either adjusting their jaw for (1) changing pitch (not a great habit, but might be necessary if it's a very wide range you're covering), (2) vowel modification (closing or opening to make that vowel easier), or (3) they've developed the habit of shaking when they add vibrato (we see Whitney Houston do that a lot).

My advice is to do what comes naturally to you! The goal is to sing quickly and cleanly, not to move the jaw - so if that helps you, fine! I just caution against relying on the jaw for every pitch change, because your jaw can't move nearly as fast as your vocal folds can.