Originating from high-fives. It's considered rude to not return a high-five when someone puts their hand in the air—there's no graceful way for them to lower it without someone returning their high-five. Their hand is left hanging in the air. So, whenever a person leaves another person in a situation where the first person is waiting for a response--and not getting it from the second person--it's referred to as being left hanging.
"Yo, dude, don't leave me hanging!"