Old English sceawian "to look at, see, gaze, behold, observe; inspect, examine; look for, choose," from Proto-Germanic *skauwojan (source also of Old Saxon skauwon "to look at," Old Frisian skawia, Dutch schouwen, Old High German scouwon "to look at"), from Proto-Germanic root *skau- "behold, look at," from PIE *skou-, variant of root *keu- "to see, observe, perceive."
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.
Show of hands is attested from 1789; Phrase for show "for appearance's sake" is from c. 1700. Show business is attested from 1850; shortened form show biz used in "Billboard" from 1942. Actor's creed the show must go on is attested from 1890. Show-stopper is from 1926; show trial first recorded 1937.