Chapter 4, "Why there is almost certainly no God" ends with a summary of what it says is "the central argument of my book". Paraphrasing, the argument is:
There is an appearance of design in nature, in both biology and physics. The biology design is just an illusion because it is best explained by Darwinian evolution. The physics design has not been explained, but there is hope that it will turn out to be a consequence of the Multiverse and the Anthropic principle.Dawkins acknowledges the multiverse is controversial, but argues that "physicists are in need of Darwinian consciousness-raising", and that "The key difference between the genuinely extravagant God hypothesis and the apparently extravagant multiverse hypothesis is one of statistical improbability".
There is no science here. The multiverse is a fantasy.
I also thought that it was odd for Dawkins to complain about "religious bigotry" later in the book. Eg, see p.287. By that he seems to me favoring one religion over another in certain circumstances. But the whole book is an attack on various religions. So his leftist politics allow him to say that all religion is bad, but do not allow others to say that particular religions are bad. Weird.