word-forming element meaning "put in or into, bring to a certain state," sometimes intensive, from French assimilation of en- "in, into" (see en- (1)) to following labial stop (-b-, -p-, and often -m-), or from the same development in later Latin in- (to im-). "This rule was not fully established in spelling before the 17th c." [OED], but it is likely the pronunciation shift was in Old French and Middle English and spelling was slow to conform. Also a living prefix in English used to form verbs from adjectives and nouns (embitter, embody). In words such as emancipate, emerge, emit, emotion the e- is a reduced form of Latin ex- (see ex-) before -m-.
Old English biter "having a harsh taste, sharp, cutting; angry, full of animosity; cruel," from Proto-Germanic *bitras- (source also of Old Saxon bittar, Old Norse bitr, Dutch bitter, Old High German bittar, German bitter, Gothic baitrs "bitter"), from suffixed form of PIE root *bheid- "to split" (source also of Old English bitan "to bite;" see bite (v.)). Evidently the meaning drifted in prehistoric times from "biting, of pungent taste," to "acrid-tasting." Used figuratively in Old English of states of mind and words. Related: Bitterly.
Others are reading
<a href="https://www.etymonline.com/word/embitter">Etymology of embitter by etymonline</a>
Harper, D. (n.d.). Etymology of embitter. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/embitter