Because of the idea contained in this http://en.wikipedia.org/wiki/Russell's_teapot To overly clarify, the burden of proof should be on the person making the statement.
We throw the word science around a lot, but this is my view of what it really is: a method for choosing what to believe. Religion is a little bit more than this. It implies that certain methods are necessary, and states what to believe. Science does not do this directly. It does not enumerate what to believe. It only states indirectly, by saying you only should believe this if it follows these guidelines.
Some people who are rightly called scientists use the "scientific method" to choose what to believe about some subset of their life, while not using the scientific method to determine what to believe about metaphysics. IE, some people are scientists and religious. And, there are some people who would split hairs about what reason is vs what science is, whatever.
The punch line is, as a method for choosing what to believe, it comes into conflict with religions, whose implied methods are mutually exclusive with science. Not that people don't try to mix the two.
Dawkin's point is that science is a much better, by large margins, method for choosing what to believe. He might say, the only correct method. This was also the subject of Carl Sagan's book The Demon-Haunted World. The difference is that Carl Sagan manages to make his point without coming off like an obnoxious twit. I do like Dawkins, but he can seem pushy and militant.
So, science doesn't have to disprove something to suggest that you shouldn't believe it. To overly simplify it, most people would say the method, science that is, says that religious people do not have valid reasons for believing what the believe. You don't have to agree with the point, that science is a much better method, to understand the motivation. Moreover, the meta point is that if all people had a much better method for choosing what to believe, then everyone would benefit. Carl Sagan makes the point, with many cases, that irrational thought, motivated by religion, caused mass suffering.
People might read his books because he could be right, that science is a much better method, and benefit as a result. Also, I believe you don't have to fully agree with someone to learn something from them. So, they might read his books to expose themselves to a wider range of philosophies. I literally die inside with disagreement with Descartes, but he was still worth reading. However, I would suggest Carl Sagan's book more than any other book in the genre that says theistic thinking is dangerous, because he is more articulate and less obnoxious.
Rush's books, on the other hand, are intellectually light polemics stuffed end-to-end with sophistry whose goal isn't to enlighten but possibly to entertain or to gratify by validating prejudices while simultaneously branding through anger so that the reader will purchase more media from Rush. I believe that Dawkins really does believe that society would be better off if they used science to decide what to believe and not faith.
At the risk of saying even more than I should say to convey my point, to put it on its head, the point is, if you didn't use science to arrive at your current belief, you shouldn't believe it.