What a terribly odd book! It's maths, but not as you know it! It's a recreational maths book on combinatorial game theory, which is about playing deterministic, total knowledge games - things like chess and go, although unsurprisingly the book focuses on things with mathematical structure which can be analysed, so it covers more things like Nim and Hackenbush.
This is volume one of four (specifically, of the second edition. The first edition was apparently in only two volumes, and printed on less annoyingly shiny paper), and concentrates on the basic, mostly adding games. When adding games, you take turns playing one move from any of the sub-games (at your choice), and then add up the scores of all the sub-games at the end. Well, it's formalised a little differently, but that's more or less the idea, and it's not unlike how the end-game of go works. (In fact Berlekamp also wrote 'Mathematical Go' on this very subject.)
Anyway, why's this book strange? First, the analysis is rather surprising, being based on Conway's 'surreal numbers' to describe game positions (rather than, say, explicit game trees), using an extended arithmetic to describe the addition of games. That's quite funky, conceptually. And the other thing is the presentation. Proofs aren't offered for most things, and indeed many ideas are used intuitively before they're formalised. For those used to other maths books, this is pretty alien. On top of this, there are plenty of silly jokes and illustrations.
In the end, I had to learn to let go and go with the flow - read ahead without thinking too hard, to see if it became clear, with a plan to read it a second time in more detail once the ideas have settled in. This is, perhaps, a good way to read maths books in general, trying to collect the overall idea first before slogging through the details. Either way, though, this is not your normal maths book!