The phrase “agent of conquest” that you use in this question is found on p. 216 in the paperback edition of the book. Diamond’s explanation for why writing is an agent of conquest is found before and after that quote. Please refer to those pages for more details with regard to this answer.
To Diamond, writing is a powerful thing. On p. 215, Diamond declares that
Knowledge is power. Hence writing brings power to modern societies by making it possible to transmit knowledge…
He goes on to say that writing is the easiest way to transmit knowledge accurately, in quantity and in detail, across space and across time.
But why is knowledge transmitted through writing important? Diamond tells us that it is important because explorers like the Spanish could write down things like the courses they sailed or the routes they explored so that later explorers could follow them. The written accounts of the riches of the Americas inspired more Spanish (and other Europeans) to come and try to take advantage. When they built empires, they administered them through writing.
All of these were things that people with writing could do but people without writing could not. They all helped to make writing an “agent of conquest.”