Apple was one of the founders of ARM. An ARM license doesn't cost them very much at all.
Manufacturing chips on the scale of Apple's iPhone means the cost per chip is relatively low. The NRE is done; at that point the more you can manufacture the cheaper it is per unit. Certainly paying Intel to manufacture chips and sell them (even at the margin that Apple can command) is going to be more expensive for Apple.
As for benefits... Apple has always wanted to own the whole shebang. They get to know ahead of time what the schedule's going to be, they get to dictate the chip's abilities, and they already have the design capability in-house. I *think* it'll be cheaper for Apple, with lower thermals and higher efficiencies with potentially a better designed chip. Whether the user sees benefits from that is up for debate.
There are certainly issues with compatibility and emulation, and I don't have a good answer for that. I suspect, if Apple go ahead and do it, they will have a good-enough answer for a transition. As for recompiling etc., they'll just require an ARM64 variant of any app in the app-store for a year or so ahead of any transition in order to be listed. That'll be sufficient IMHO to get almost everyone on-board.