To match across a distance — one thing answering another's shape; and to exchange letters, which is the same thing between people.
Medieval Latin correspondere (to answer to, match), from com- (together, with) + respondere (to answer), from re- (back) + spondere (to pledge); one thing answering another's shape, extended to people answering each other by letter.
Latin spondere is to pledge, and an answer given this way was once a promise, not just a reply — respondere carried an obligation, and correspond keeps that weight across a distance. Letters exchanged this way were always more binding than mere conversation.