The short answer to this question is that Europeans brought many diseases to which Native Americans had no immunity, and that Native Americans had few diseases that had similar effects on Europeans. But the deeper answer is that Europeans had developed a number of diseases that had evolved as epidemic, or crowd, diseases. This was because Europe featured a much denser population than the New World, which was in turn due to the earlier development of agriculture, which happened because of a number of geographic advantages. While Diamond admits that Native Americans had many dense populations, (he cites an estimate that about 20 million Indians lived in the Americas before Columbus arrived) he also observes that sedentary, dense societies developed later in the Americas, thus giving epidemic diseases less time to evolve. Additionally, many of the terrible epidemic diseases that Europeans developed had their origins in domesticated animals that Indians did not have. So, simply stated, Europeans just brought more diseases than Indians had.
Source: Diamond, Guns, Germs, and Steel, 195-214.