The author sets out to create what he calls a "third category" of mathematics book--neither stories about mathematics nor technical books for mathematicians, but real mathematics for a lay audience. This is a strange thing to write, because there are many excellent books of this type.
This one concentrates on two popular recreational mathematics topics, set theory and number theory, in eight chapters (called "lessons") all involving infinity in some manner. Compared to other recreational mathematics books, the presentation is more formal and academic than average, but still entirely accessible to readers without mathematical training beyond middle school. It's also drier than most with examples more textbook-like than fanciful. The writing is clear and pleasant, but not humorous or stylish.
I think the best audience for this book is someone who would like to learn some number theory and set theory without tackling a mathematics text. It's non-technical and entertaining enough to be read for pleasure while still learning a lot.
There are much more entertaining recreational books on similar material. I would recommend those for readers who value entertainment over learning, or those who will follow up the book with deeper study in the field. This book is a painless, even pleasant, way to learn the basics of the fields.