I can't speak for the book, but I think that Agile Software Development reduces costs in a number of ways:
I agree with Per that some of the practices directly reduce costs.
Test Driven Development, for example, radically reduces the number of bugs you introduce in the software, and thereby significantly reduces the amount of time you need to spend debugging.
Additionally, being quite humane approaches to software development, Agile approaches often lead to better motivated teams. I hope it's no secret that better motivated teams are more effective.
Last but not least, Agile practices do away with a lot of waste in software development. And I'm not only talking about tons of documents that are created but never used.
It is known that typically more than half the features of a software system are never or seldom used. It's not hard to imagine that for most of those it would have saved costs to not implement them.
Why are they implemented? Mostly, I think, because people don't know what they need, let alone what they will need in, say, a year from now. But when we tell them that they need to come up with their requirements now, and that it will be costly to change them later, we force them to come up with all those features they actually aren't sure they will really need, just in case.
Agile Software Development approaches change that - we only develop the set of most important features and then show them to the users to try them. Then they can learn from that experience and decide what else they need. We tell them that, as long as a feature isn't yet implemented, changing it or exchanging it for something totally different costs near to nothing. That way we dramatically improve the probability that the final system will really contain all important features, and we don't waste much time on features that aren't of use to anyone.
One more thing that doesn't reduce cost directly, but increases income, is that Agile teams work on having a releasable system from early on. If a system with the most important features implemented can be installed and used in a couple of weeks instead of months or years, we get earlier and more return for our investment.