Carse's book is so aphoristically written that it's polarizing. Some people think it's genius, some people think it's nonsense, some people feel there's something deep behind it but can't put their finger on it and figure maybe they're just not smart enough to get it.
The book is basically about competition (finite games) vs cooperation (infinite games), written in aphorisms without any mathematics or reference to game theory. Carse is a theologist, not a mathematician, so I'm guessing he discovered patterns of interaction that he didn't have the mathematical tools to explain and this is the result - a long list of fortune cookie style descriptions.
If you want to get the math that drives it, look up game theory, in particular Axelrod's work on Cooperation. Axelrod found that lengthening time in the form of allowing multiple iterations (which in Carse's terms means going from finite to infinite), causes people to cooperate and pursue win/win situations instead of the standard win/lose inherent in competition.
The genius in this as well as Carse's descriptions is that every interaction can fall into one of these categories and there are a ton of implications for adopting the wrong approach. This is especially relevant in the age of tribalism when we have politicians/groups/countries who are in it to win at any cost, even at the expense of burning bridges with people/groups/countries they're still gonna have to deal with in the future. Maybe they should have played the Infinite Game instead of the Finite one?
Anyway, if you read this, just think competition vs cooperation (game theory) instead of finite vs infinite and everything will start to make sense. And to quote Venkat Rao, this book pays back your thinking many times fold
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