1520s, "to be in agreement, to be in harmony with," from Middle French correspondre (14c.) or directly from Medieval Latin correspondere, from cor- (see com-) "together, with each other" + respondere "to answer" (see respond).
Originally in Medieval Latin of two things in mutual action, but by later Medieval Latin it could be used of one thing only. In English, sense of "to be similar" (to) is from 1640s; that of "to hold communication with" is from c.1600; specifically "to communicate by means of letters" from 1640s (in mid-18c. it also could mean "have sex"). Related: Corresponded; corresponding.