by 1862, in sporting jargon (curling), "to play (a match) again," from re- "again" + play (v.). Of sound recordings (later video, etc.), "reproduce what has been recorded," by 1912. Related: Replayed; replaying.
The noun is from 1895 as "a replayed match" in sports. The meaning "action of replaying" a sound recording, film, later also video, etc., is by 1953.