There are a good number of etymological myths in the English language, stories about the origins of words (such as the widespread notion that posh originated as an acronym for “port out, starboard home”) which are, to put it kindly, inaccurate. But this does not mean that every vivid account of linguistic origin is fictitious. Many words, such as fan, do have colorful backstories.
Fan is generally–and very likely correctly–believed to be a shortened form of fanatic. The origin of fanatic (which can be traced back to the Latin word fanum, meaning “sanctuary, temple”) is less often commented on. In English, fan made an early appearance in the late 17th century only to disappear for two centuries, resurfacing in the late 19th century. In this later period of use, it often referred to the devoted observers of, or participants in, a sport. An 1885 article from The Kansas City Times, for example, contains the line “The base ball ‘fans’ of the ploice [sic] force and fire department engage in a ball game.”