The reason for this is the fact that the people of the Old World had had domesticated animals in larger numbers and for longer times than the people of the New World had.
Diamond tells us that the epidemic diseases that killed so many of the people of the New World were passed to people from the animals that they domesticated. This means that people who had more domesticated animals, and who had them longer, were more likely to have had the time and the opportunity to develop epidemic diseases. It was the people of the Old World who were able (for geographic reasons) to domesticate animals first and to have more kinds of domesticated animals. For this reason, it was they who got the "germs" that helped them conquer the New World.