His central argument in his book "The God Delusion".
I don't have a copy of 'The God Delusion', so I can't verify that the words that you quoted are Dawkins' or that they come from his book. I do notice that this text is found on many Christian websites.
1. One of the greatest challenges to the human intellect, over the centuries, has been to explain how the complex, improbable appearance of design in the universe arises.
It's hard to argue with that one.
2. The natural temptation is to attribute the appearance of design to actual design itself. In the case of a man-made artifact such as a watch, the designer really was an intelligent engineer. It is tempting to apply the same logic to an eye or a wing, a spider or a person.
In real life we observe that some instances of functional form are the intentional designs created by an artisan, while others (typically biological form) seem to occur naturally.
The 'design argument' is essentially an analogy, in which one treats all instances of functional form as instances of intentional design.
This analogy once had great persuasive force because of the difficulty that people experienced in understanding how the functional form could have come about naturally, without some invisible artisan having designed it in.
3. The temptation is a false one, because the designer hypothesis immediately raises the larger problem of who designed the designer. The whole problem we started out with was the problem of explaining statistical improbability. It is obviously no solution to postulate something even more improbable. We need a "crane", not a "skyhook", for only a crane can do the business of working up gradually and plausibly from simplicity to otherwise improbable complexity.
We started out with the problem of accounting for the functional structure observed in nature. Attributing it to a designer leaves us with a new and different problem, namely understanding and accounting for the designer.
Explantions produce understanding of how and why things are as they are. That means that we need to know more after we heard the explanation then we knew before.
If we are going to imagine a supernatural designer, one that operates through inexplicable miracles, then we seem to be generating mystification instead of explanation.
4. The most ingenious and powerful crane so far discovered is Darwinian evolution. Darwin and his successors have shown how living creatures, with their spectacular statistical improbability and appearance of design, have evolved by slow, gradual degrees from simple beginnings. We can safely say that the illusion of design in living creatures is just that – an illusion.
Natural selection provides an alternative naturalistic explanation of how functional form came about that doesn't depend on miraculous interventions by inexplicable invisible designers.
5. We don't yet have an equivalent "crane" for physics. Some kind of multiverse theory could in principle do for physics the same explanatory work as Darwinism does for biology. This kind of explanation is superficially less satisfying than the biological version of Darwinism, because it makes heavier demands on luck. But the anthropic principle entitles us to postulate far more luck than our limited human intuition is comfortable with.
6. We should not give up the hope of a well-grounded explanatory model arising in physics, something as powerful as Darwinism is for biology. But even in the absence of a strongly satisfying model to match the biological one, the relatively weak models we have at present are, when abetted by the anthropic principle, self-evidently better than the self-defeating God hypothesis of an intelligent designer.
The newer anthropic design arguments insist that the universe is "fine tuned" to be hospitable to human life. Of course, if human beings evolved in this universe, then we would expect humans to be adapted to the universe that they evolved in.
If physical principles were significantly different than those of our universe, and if cognitive beings evolved in those conditions, then those cognitive beings might generate similar arguments about how uniquely suited their universe is to them
7. If the argument of this chapter (book) is accepted, the factual premise of religion -- the God hypothesis – is untenable. God almost certainly does not exist. This is the main conclusion of the book so far.
That's too strong. The numbered assertions quoted above don't necessarily make "the God hypothesis" untenable in all contexts. Nor, for that matter, is "God" the "factual premise of religion".
But the points quoted above do suggest some of the reasons why the theistic design arguments aren't really convincing.