。
She bowed in response to their applause.
Literal
She [topic-は] their [genitive-の] applause [to-に] responding bow-did.
~に答える ('respond to, answer (to)') uses に to mark what's being responded to. お辞儀する is the standard verb for the Japanese bow — お is a fossilized politeness prefix that's now an inseparable part of the word. Bowing serves many functions in Japanese: greeting, thanking, apologizing, acknowledging applause. The te-form 答えて links cause (acknowledging the applause) to response (bowing).