Why does Moriarty owe Sherlock a fall? I mean why did he specifically emphase that he owed Sherlock a fall? What happened between them?
Source Quora :
Moriarty promised way back at the swimming pool that he was going to kill Sherlock eventually. And not only kill him, but burn him completely in the process, if Sherlock didn't back off and leave him alone.
"I owe you" is simply a reminder of that threat. Similarly, the final problem, the one about which Moriarty says, "I told you, but did you listen" is, literally, Stayin' Alive, as Moriarty says when Sherlock comes to find him on the St. Bart's roof. Sherlock's the only thing that makes Moriarty's life interesting, but he can't let Sherlock live because he'll keep interfering.
Of course, there's also the theory that Jim was planning the Fall way back from the very, very beginning, right down to planning the precise moment he'd get the phone call from Irene Adler at the swimming pool that resolved his standoff with Sherlock. More...