How Does Aizawa Control His Ropes?

Aizawa controls the binding cloth through static energy via levitation. If he activates his Erasure Quirk, the cloth swirls; his tangled jet-black hair also stands up! Also note that the cloth floats midair; that is, about 12-inches at a time, and not altogether.

Why does Mr Aizawa's scarf float?

Exactly! A mini-antigravity on himself, it just happens to affect his hair AND his scarf. TL;DR - When Aizawa activates his quirk it makes things near his head float, including his hair and scarf.

What are Aizawa's wraps called?

The Capturing Weapon (捕縛武器, Hobaku Buki^?) is a weapon used by Shota Aizawa and Hitoshi Shinso.