What does 回, the kanji for "turn", mean in 回答 "answer"?
In Chinese, this kanji (or hanzi) originally contains a notion of "returning" (both transitive and intransitive); for example, 回来 "to come back", 回家 "to go back home", etc.
Although in Japan this kanji seems to mean more of "rotating" than "returning", in this case, the original meaning has been carried over into Japanese.
So 回答 is about "returning an answer."
[Edit] More precise answer is provided below by Broccoli forest. Please don't miss it!