Old English ræcan, reccan "to reach out, stretch out, to extend, hold forth," also "to succeed in touching, succeed in striking; to address, speak to," also "to offer, present, give, grant," from Proto-West Germanic *raikejanan "stretch out the hand" (source also of Old Frisian reka "to give, pay," Middle Dutch reken, reiken, Old High German reihhen, reichen "give, reach out, get," Dutch reiken, German reichen "to reach, to pass, to hand, to give; to be sufficient"), from Proto-Germanic *raikijanau, which is probably from PIE root *reig- "to stretch, stretch out, be stretched; be stiff."
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.