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.
c. 1300, "act of exhibiting to view," from show (v.). Sense of "appearance put on with intention to deceive" is recorded from 1520s. Meaning "display, spectacle" is first recorded 1560s; that of "ostentatious display" is from 1713 (showy is from 1712). Sense of "entertainment program on radio or TV" is first recorded 1932. Meaning "third place in a horse race" is from 1925, American English (see the verb).
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 magazine from 1942. Actor's creed the show must go on is attested from 1890. Show-stopper is from 1926; show trial is attested by 1937.