fan (n.1) Look up fan at Dictionary.com
device to make an air current, Old English fann (West Saxon) "a basket or shovel for winnowing grain" (by tossing it in the air), from Latin vannus, perhaps related to ventus "wind" (see wind (n.1)), or from PIE root *wet- (1) "to blow" (also "to inspire, spiritually arouse;" see wood (adj.)).

The chaff, being lighter, would blow off. Sense of "device for moving air" first recorded late 14c.; the hand-held version is first attested 1550s. A fan-light (1819) was shaped like a lady's fan. The automobile's fan-belt is from 1909. Fan-dance is from 1872 in a Japanese context; by 1937 as a type of burlesque performance.
fan (n.2) Look up fan at Dictionary.com
"devotee," 1889, American English, originally of baseball enthusiasts, probably a shortening of fanatic, but it may be influenced by the fancy, a collective term for followers of a certain hobby or sport (especially boxing); see fancy (n.). There is an isolated use from 1682, but the modern word likely is a late 19c. formation. Fan mail attested from 1920, in a Hollywood context; Fan club attested by 1930.
fan (v.) Look up fan at Dictionary.com
late Old English fannian "to winnow (grain)," from the noun (see fan (n.1)). Meaning "to stir up air" is from early 15c. Baseball sense of "strike out (a batter)" is by 1909. Related: Fanned; fanning. To fan out "spread out like a hand-held fan," is from 1590s.