late 15c., "to divert the attention, beguile, delude," from Old French amuser "fool, tease, hoax, entrap; make fun of," literally "cause to muse" (as a distraction), from a "at, to" (from Latin ad, but here probably a causal prefix) + muser "ponder, stare fixedly" (see muse (v.)).
The original English senses are obsolete; the meaning "divert from serious business, tickle the fancy of" is recorded from 1630s, but through 18c. the primary meaning was "deceive, cheat" by first occupying the attention. "The word was not in reg. use bef. 1600, and was not used by Shakespere" [OED]. Bemuse retains more of the original meaning. Greek amousos meant "without Muses," hence "uneducated."