Old English andswaru "a response, a reply to a question," from and- "against" (from PIE root *ant- "front, forehead," with derivatives meaning "in front of, before") + -swaru "affirmation," from swerian "to swear" (see swear), suggesting an original sense of "sworn statement rebutting a charge." Meaning "solution of a problem" is from c. 1300.
It is remarkable that the Latin expression for answer is formed in exactly the same way from a verb spondere, signifying to engage for, to assure. [Hensleigh Wedgwood, "A Dictionary of English Etymology," 1859]
A common Germanic compound (cognates: Old Saxon antswor, Old Norse andsvar, Old Frisian ondser, Danish and Swedish ansvar), implying a Proto-Germanic *andswara-. The simpler idea of "a word in reply" is expressed in Gothic anda-vaurd, German Antwort.
Old English answarian "make a statement in reply," from and- "against" (from PIE root *ant- "front, forehead," with derivatives meaning "in front of, before") + swerian "to swear" (see swear), suggesting an original sense of "make a sworn statement rebutting a charge."
Meanings "conform, correspond" and "be responsible for" are from early 13c; that of "suffer the consequences of" is from late 13c.; that of "respond in antiphony" is from early 15c. Sense of "respond in act or action, give back in kind" is from 1570s; that of "solve, find the result of" is from 1742. Related: Answered; answering. The telephone answering machine so called from 1961.