The answer to this question can be found in Chapter 9. Specifically, the answer begins on p. 161 of the paperback edition of the book.
In this book, Diamond’s major thesis is that some regions of the world became more powerful than others simply because of luck. The Europeans, he says, were able to conquer people in the Americas, Australia, and elsewhere because they were lucky in terms of their geography. One of the ways in which the Europeans were lucky is in the fact that they lived in a place with many animals that could be domesticated.
Diamond says that there were fourteen big, herbivorous mammals that were domesticated in ancient times. Of these, thirteen were from Eurasia. On p. 162, he says that the main reason for this was that
Eurasia has the largest number of big terrestrial wild mammal species, whether or not ancestral to a domesticated species.
In Table 9.2, he points out that Eurasia had 72 species that could possibly have been domesticated. This is almost as many as all of the rest of the world combined. He then goes on to argue that almost all of the animals in other parts of the world were (and still are) impossible to domesticate.
Thus, Eurasians domesticated so many more animals than other people because they had more animal species to begin with and their species were more amenable to domestication.