1530s (transitive), "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, destroy the repute of" (a sense surviving in an exploded theory), from ex "out" (see ex-) + plaudere "to clap the hands, applaud," which is 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.
At the close of the performance of a comedy in the Roman theatre one of the actors dismissed the audience, with a request for their approbation, the expression being usually plaudite, vos plaudite, or vos valete et plaudite. [William Smith, "A First Latin Reading Book," 1890]
English used it to mean "drive out with violence and sudden noise" (1650s), later "cause to burst suddenly and noisily" (1794). Intransitive sense of "go off with a loud noise" is from 1790, American English; figurative sense of "to burst with destructive force" is by 1882; that of "burst into sudden activity" is from 1817; of population by 1959. Related: Exploded; exploding.