late 14c., "to say what one has already said," from Old French repeter "say or do again, get back, demand the return of" (13c., Modern French répéeter), from Latin repetere "do or say again; attack again," from re- "again" (see re-) + petere "to go to; attack; strive after; ask for, beseech" (from PIE root *pet- "to rush, to fly"*).
Meaning "say what another has said" is from 1590s. As an emphatic word in radio broadcasts, 1938. Meaning "do over again" is from 1550s; specific meaning "to take a course of education over again" is recorded from 1945, American English. Related: Repeated; repeating.
* pet- "to rush, to fly"