The answer to this can be found in Chapter 14. Specifically, it can be found on page 285 of the paperback edition. On that page, Diamond gives us three reasons why food production is necessary in order for a state to come about.
The first reason is that farming “involves seasonally pulsed inputs of labor.” What that means is that farmers do not have to work as much at certain times of year. They have to work hard at planting and at harvest, but at other times of the year the central government can require them to do work for the government.
The second reason is that food production can lead to
…stored food surpluses, which permit economic specialization and social stratification.
If you do not have food production, everyone has to hunt and gather. This means that there is no time for some people to become weavers and others to become metal workers. It also means that everyone has about the same amount of possessions so there can’t be rich and poor people. Once you have food production, you can “feed all tiers of a complex society.”
Finally, if people engage in food production, they have to live sedentary lives. If they live in one place all the time, they can accumulate wealth. They can be called on to work for the political and religious leaders of the society. This is not possible if people are moving around nomadically.
These are Diamond’s three explanations for why food production is necessary for state-building.