I enjoyed this, but like Scattercat, I found it difficult to see why the game was so compelling in the end. I don't think it's a matter of asking why - ThinlyVeiledAlias is correct that why they play to begin with doesn't matter so much. Rather, it's the rather bizzare shift in Danny's attitude at the end.
The story built up suspense slowly and masterfully, and it accumulated and accumulated and then the reveal was at just the right time. I, as a listener, and Danny, as protagonist, are shocked by the surgery. Then Danny is forced to go through with it, and I feel for him. And then Danny leaves, shocked, trying to adjust, and trying to figure out a way to survive the next game - and at the time, the feeling is that he would happily avoid it, he just doesn't think they'll let him. So, ok - I don't know how it feels like to have an extra hand, but I understand what it is like to feel trapped - so I am still following along with Danny here. Then, Danny has a clever idea! Amoral, sure, but in character and I'm happy to play along. And now Danny is hooked and wants to keep playing and I have no idea how that happened.
That's my problem with this story - not that the game doesn't make sense, but that Danny went from being terrified to being eager without even playing a second time. If the story had cut forward a few games in the future, and now Danny is hooked - I would have had no issue with it. But basically, for a story that was so good at keeping not only my interest but my empathy throughout, the ending was too much and broke my connection to it.