。
She graciously agreed to my request.
Literal
She [topic-は] willingly my request [to-に] responded [for-me-てくれた].
快く is the adverbial form of 快い ('pleasant, agreeable') and means 'willingly, readily, with good grace' — the opposite of grudgingly. 要求に応じる is a standard collocation, '~に応じる' meaning 'to respond/comply with ~.' Tacking on ~てくれる ('do for me / do for my benefit') signals that the speaker is the grateful recipient of her good grace; this is the giving-and-receiving (授受) layer of Japanese verbs that learners often gloss over but native speakers reach for routinely to mark social dynamics.