Dawkins starts off the book by trying to clear up the common misunderstanding of what a scientific "theory" means. In short, it is an explanation of a natural phenomenon that has been rigorously tested and verified using the scientific method as opposed to the everyday meaning of the word which is more akin to a scientific “hypothesis”. Thus, to call evolution a theory is not to undermine it but to give it the highest possible credibility.
Dawkins then provides a gentle introduction to the theory of evolution by first looking at selective breeding and using that as a segway to introduce natural selection. He then dedicates the rest of the book to present all sorts of evidence from different branches of science such as the fossil record, comparative anatomy, biogeography (geographical distribution of organisms), direct observation, and molecular biology.
Dawkins demonstrates how, like any good scientific theory, evolution is falsifiable (vulnerable to disproof) in a number of ways, but that no such attempts have so far succeeded, despite the desperate efforts of many. For example, evolution would be dismantled by the discovery of a single fossil in the wrong geological stratum (e.g., a rabbit fossil in the Precambrian).
The book also deals with some of the usual creationist challenges, including the so-called missing links, as well as discussing common misconceptions of evolution and how these can lead to misunderstandings (e.g., a failure to appreciate the huge time spans that evolution operates in).
The book is written in an accessible way, meaning that even laypeople like me are able to follow most of the text, although some parts do get a little technical but perhaps inevitably so for such a complex subject.
In short, Dawkins does an excellent job of giving an accessible overview of how evolution by natural selection works and the mountain of evidence that makes it a scientific fact (yes, a scientific theory can also be a fact).
Don’t expect this book will tell you everything you need to know about evolution, especially if, like me, you do not have a science background. But nevertheless, it is a very good introductory book and I can now press on with further studying!