The answer to this can be found in Chapter 8 of the book.
Basically, as with all of Diamond's arguments, the idea is that the Agricultural Revolution started in the Middle East because of geographical luck. Diamond argues that this area had more species of plants and animals that could be domesticated. Since there were so many of these species, it was much easier for the people of this area to start to farm. In addition, Diamond argues that the species of plants in the Middle East were much better in terms of their nutritional value and in terms of how easy it was to domesticate and raise them.
Because of these factors, which had only to do with luck, the people of the Fertile Crescent developed agriculture first.