The coat closet is at the front of the aircraft. You aren't really allowed to just go rummage through it yourself. So you ask the FA to put your coat in there or to retrieve it.
Ok, as a guy that only flies coach I thought "hanging your coat" means putting it on the little peg that pops out on the back of the seat in front of you / putting it in the overhead bin. I was mistaken. I didn't realize coats at the front of the plane were treated so much better than the rest of the coats. :-)
They'll also memorize your name and make sure to use it regularly when addressing you. And you get metal cutlery instead of plastic. And the nuts are heated, and served in a little cup instead of in a plastic bag.
(Never paid for business, but United is ludicrously easy to get upgrades on if you fly trans-atlantic flights with any frequency and pay attention to ordering correctly coded tickets)
The coat closet is at the front of the aircraft. You aren't really allowed to just go rummage through it yourself. So you ask the FA to put your coat in there or to retrieve it.