1530s, "to reject with scorn," from Latin explodere "drive out or off by clapping, hiss off, hoot off," originally theatrical, "to drive an actor off the stage by making noise," hence "drive out, reject" (a sense surviving in an exploded theory), from ex- "out" (see ex-) + plaudere "to clap the hands, applaud," of uncertain origin. Athenian audiences were highly demonstrative. clapping and shouting approval, stamping, hissing, and hooting for disapproval. The Romans seem to have done likewise.
English used it to mean "drive out with violence and sudden noise" (1650s), later, "go off with a loud noise" (American English, 1790); sense of "to burst with destructive force" is first recorded 1882; of population, 1959. Related: Exploded; exploding.