It's one of those cliches that you never really think about, and just assume that "it's always been this way". But there's no obvious reason why spirits and ghosts should look mostly cloth-like with convenient eye-holes à la Scooby Doo.
So where did that image come from, originally? Dickens illustrations? King Lear? Because I have no idea.
Good question! My understanding is that the "bedsheet ghost" originally developed from the idea of the deceased appearing as they did at the time of burial, dressed in a funeral shroud. Up until sometime in the 19th century (and still today for Jews and Muslims), most European Christians were buried wrapped only in a simple white sheet.
However, probably more influential is the practice of using a white shroud as a short-hand for "ghost" in 19th century theater performances, especially of "Hamlet" and other plays where the ghosts need to be visually distinguishable from the other living characters.
hi.. you may be interested in a couple of earlier posts on ghosts. They don't address the question of the 'bedsheets with eye-holes' image specifically, but may provide some background
Did the Romans or other ancient civilizations ever report paranormal phenomena such as ghosts, aliens, UFOs, etc? - an early reference to ghosts being white
Did ghosts become white in America and the rest of the west because of slavery? - more on white, and some discussion of shrouds
edit: oh, and this one may be interesting too
