The answer to this question can be found in Chapter 12. We must realize, however, that there is no way for Diamond or anyone else to know for certain why kings thousands of years ago did something. We can really only speculate. Diamond’s speculation is that the kings limited the knowledge of writing to a few of the elites because they did not want writing to be used by other people to do things that might reduce the kings’ power.
In Chapter 12, Diamond quotes the anthropologist Claude Levi-Strauss. On p. 235, he quotes Levi-Strauss as saying that the main point of writing in the ancient world was "to facilitate the enslavement of other human beings."
The idea here was that the kings wanted to use writing for their own purposes. They wanted to use it to keep track of who owned how much so they could know how much everyone had to pay in the way of taxes. They did not want other people to learn how to write because other people might use writing to do things like plotting against them.
Basically, then, Diamond says that kings wanted to limit literacy so as to maintain more power for themselves.