Old English þrote (implied in þrotbolla "the Adam's apple, larynx," literally "throat boll"), related to þrutian "to swell," from Proto-Germanic *thrut- (cf. Old High German drozza, German Drossel, Old Saxon strota, Middle Dutch strote, Dutch strot "throat"), perhaps from PIE *trud- (cf. Old English þrutian "to swell," Old Norse þrutna "to swell").
The notion is of "the swollen part" of the neck. Italian strozza "throat," strozzare "to strangle" are Germanic loan-words. College slang for "competitive student" is 1970s, from cutthroat.