Old English sceawian "to look at, see, gaze, behold, observe; inspect, examine; look for, choose," from West Germanic *skauwojan (cf. Old Saxon skauwon "to look at," Old Frisian skawia, Dutch schouwen, Old High German scouwon "to look at;" Dutch schoon, Gothic skaunjai "beautiful," originally "conspicuous"), from Proto-Germanic root *skau- "behold, look at," from PIE *skou-, variant of root *skeue- "to pay attention, perceive" (see caveat).
Causal meaning "let be seen; put in sight, make known" evolved c.1200 for unknown reasons and is unique to English (German schauen still means "look at"). Spelling shew, popular 18c. and surviving into early 19c., represents obsolete pronunciation (rhymes with view). Horse racing sense is from 1903, perhaps from an earlier sense in card-playing.