Although most of the chapter is devoted to talking about how people developed writing systems, there is a bit at the beginning that tells you why writing systems are important to a civilization (look in the second and third paragraphs of the chapter). Diamond summarizes the idea by saying "Knowledge brings power."
What Diamond is saying there is that knowledge is important and that writing can bring (and preserve) more knowledge than is possible without writing. When a civilization is able to write, they can record information about how they have done things. These records can be exact and can last. Civilizations can also get ideas from faraway places by writing them down.
The basic idea, then, is that a civilization with writing can preserve its knowledge by writing it down. It can also gather more knowledge than it could without writing. By having more knowledge, it can have more power.
The importance Diamond assigns to writing is succinctly described in the following passage from Chapter 12, entitled "Blueprints and Borrowed Letters":
Knowledge brings power. Hence writing brings power to modern societies, by making it possible to transmit knowledge with far greater accuracy and in far greater detail, from more distant lands and more distant times...Writing marched together with weapons, microbes, and centralized political organizations as a modern agent of conquest.
Writing enabled the conquerors to record information about the lands and people they encountered, which facilitated their rapid conquest. Diamond goes on to observe that writing was enormously difficult to invent, and spread largely through diffusion. Its main use was for record-keeping, which accompanied and facilitated the development of complex bureaucracies. As such, it remained the province mostly of elites until very recently in world history. In any case, the fact that writing, like other important technologies, spread through Eurasia relatively quickly, and not to other regions, was a result, Diamond argues, of geographic factors instead of cultural ones.
Source: Jared Diamond, Guns, Germs, and Steel: The Fates of Human Socieites (New York: W.W. Norton and Company, 1999), 215-216.