"Passed on" means to decline an offer or an opportunity. In this case the author is thinking about the cheeseburgers he'd had the chance to eat earlier but chose not to. On is being used here as part of a phrasal verb like, "to run along" or, "to buy out." It doesn't mean anything on its own. Phrasal verbs are tricky but keep practicing and I'm sure you'll pick them up in no time!

It's an inseparable phrasal verb - 'to pass on' something. If you 'pass on' a cheeseburger, you don't accept one when it's offered to you. For example. 'Do you want a cheeseburger?' 'No thanks. I'll pass on that. I'm not very hungry.' It's similar to saying 'pass' when you are doing a quiz and you're asked a question which you don't want to answer.

"on" is short for "on to someone else" but we take out "to someone else" to make it shorter (so the narrator gave a cheeseburger to someone)