A person who has been "left hanging" has been told a story, but denied any resolution (if it's a joke, they don't get to hear the punchline. For example:

"So he said, 'Boy, if you touch my dog again I'll rip your throat out,' and I said 'dog? I thought she was your daughter!" He bared his teeth and... oops, I'm late for work!"
"What? You can't just leave me hanging!"

The phrase isn't meant to invoke hanging as a method of execution, but rather the idea of being helplessly suspended. cf cliffhanger.