Прогибаться oten means to yield or to please. It is very informal when spoken of people as opposed to the physical/mechanical meaning of the word.
A few samples here:

Woman-Boss: We're working tonight till 12pm.
Manager: Sure thing, boss. We'll be happy to be looking at the beauty of yours infinitely.
Boss chuckles and goes away trying to suppress victorious smile
A co-worker hisses: "прогиб засчитан" meaning she has appreciated your flattery
A talk of a top-manager to boss after a meeting regarding reconsideration of anything (deal details, delivery terms, etc):
Boss: Why? Why are you coming here to say that we have to wait those additional four weeks?
Man: I'm sorry, boss. I was trying my best but they said they had a super-massive prepaid order so they could sacrifice everything else in order to fulfill that one... They said ours would be the very-first after that one. They were so persuasive that I had to yield (Они были так убедительны, что пришлось прогнуться (уступить)...