c. 1300, "falsehood, fictitious narrative; a lie, pretense," from Old French fable "story, fable, tale; drama, play, fiction; lie, falsehood" (12c.), from Latin fabula "story, story with a lesson, tale, narrative, account; the common talk, news," literally "that which is told," from fari "speak, tell" (from PIE root *bha- (2) "to speak, tell, say").
The restricted sense of "animal story" (early 14c.) comes from the popularity of Aesop's tales. In modern terms, "a short, comic tale making a moral point about human nature, usually through animal characters behaving in human ways" ["Oxford Dictionary of English Folklore"].
The fable which is naturally and truly composed, so as to satisfy the imagination, ere it addresses the understanding, beautiful though strange as a wild-flower, is to the wise man an apothegm, and admits of his most generous interpretation. When we read that Bacchus made the Tyrrhenian mariners mad, so that they leapt into the sea, mistaking it for a meadow full of flowers, and so became dolphins, we are not concerned about the historical truth of this, but rather a higher poetical truth. We seem to hear the music of a thought, and care not if the understanding be not gratified. [Thoreau, "A Week on the Concord and Merrimack Rivers"]