The answer to this question can be found in Chapter 9. There, Diamond tells us that the people of Eurasia domesticated more animal species simply because they were lucky. They had more kinds of animals that could be domesticated. This was not due to any virtue on their part; it was simply due to luck.
Diamond’s main argument in this book is that it was geographical luck that made some regions of the world successful and others weaker and less powerful. It was not that the people of Eurasia were racially or culturally better. Instead, they were simply luckier.
One aspect of this luck was that they had more animals that could be domesticated. Diamond argues that there are very few large animals in the world that can really be domesticated effectively. Too many animals have something “wrong” with them. For example, they are too mean or they are carnivorous or they grow too slowly. Diamond points out that Europeans and their descendants have not been able to domesticate animals like kangaroos or zebras any more than the natives of Australia and Africa were. This means that those animals were simply not domesticable and the people of Australia and Africa were unlucky. By contrast, there were many large, domesticable animals in Eurasia.
Thus, Diamond argues that the people of Eurasia domesticated more species of large animals simply because they were lucky enough to have many suitable animals available to them.