Looking at the multitude of meanings for the word fall, one thing comes clear: the word is most often associated with a drop or descent of some kind, whether that be a physical fall from a high place or a metaphorical fall from power or grace.
In the 16th Century, autumn was known as "the fall of the leaf," which later became just fall, but the actual word is much older than that, traceable to the Old English feallan. Many meanings, including a fall in temperature, to fall in love or to fall asleep, all stem from the Middle Ages. If you're the "fall guy," you're going to get blamed, or "take the fall." If you fall in love and then fall out of favor with your beloved, your next action might fall under the category of "gift giving" or just "kissing up."