Old English etan (class V strong verb; past tense æt, past participle eten) "to eat, devour, consume," from Proto-Germanic *etanan (cf. Old Frisian ita, Old Saxon etan, Middle Dutch eten, Dutch eten, Old High German ezzan, German essen, Old Norse eta, Gothic itan), from PIE root *ed- "to eat" (see edible).
Transferred sense of "slow, gradual corrosion or destruction" is from 1550s. Meaning "to preoccupy, engross" (as in what's eating you?) first recorded 1893. Slang sexual sense of "do cunnilingus on" is first recorded 1927.