1530s, "to cause to happen sooner," a back-formation from anticipation, or else from Latin anticipatus, past participle of anticipare "take (care of) ahead of time," literally "taking into possession beforehand," from anti, an old form of ante "before" (from PIE root *ant- "front, forehead," with derivatives meaning "in front of, before") + capere "to take," from PIE root *kap- "to grasp."
Later "prevent or preclude by prior action" (c. 1600) and "be aware of (something) coming at a future time" (1640s). Used in the sense of "expect, look forward to" since 1749, but anticipate has an element of "prepare for, forestall" that, etymologically, should prevent its being used as a synonym for expect. Related: Anticipated; anticipating.