Old English ræcan "to extend, hold forth," also "to succeed in touching," from West Germanic *raikjan "stretch out the hand" (cf. Old Frisian reka, Middle Dutch reiken), from Proto-Germanic *raikijanau, perhaps from PIE root *reig- "to stretch out" (cf. Sanskrit rjyati "he stretches himself," riag "torture" (by racking); Greek oregein "to reach, extend;" Lithuanian raizius "to stretch oneself;" Old Irish rigim "I stretch"), related to base *reg- "to rule, to lead straight, to put right" (see regal).
Shakespeare uses the now-obsolete past tense form raught (Old English ræhte). Meaning "arrive at" is early 14c.; that of "succeed in influencing" is from 1660s. Related: Reached; reaching. Reach-me-down "ready-made" (of clothes) is recorded from 1862, from notion of being on the rack in a finished state.