Diamond argues that the move from animals to humans came in four stages. This starts on p. 207 in my book -- about 14 pages in to the chapter.
First, there are diseases that come directly to people from animals. This can happen, for example, if a microbe enters a person as they butcher an animal.
Second, the animal pathogen can evolve and then get transmitted. Diamond says that in this stage, the epidemic dies out for one reason or another. An example of this was kuru, which ended when cannibalism in New Guinea ended.
Third, you can have pathogens that evolve, cross over, and do not die out. That seems to be what has happened with AIDS.
Fourth, there are microbes that have moved over to us and that cannot make it back to animals.
The ancestors of microbes now confined to humans transferred themselves from animal to human hosts through the process of evolution. A microbe that relies on what the author calls one "arthropod vector" can switch to a new arthropod vector if the microbe has found a new host. For example, the disease typhus was originally spread among rats by rat fleas and through rat fleas to humans. Then, typhus microbes began traveling on body lice as a more efficient means of infecting humans directly. Now that delousing has eliminated this route of transmission, typhus is transmitted through North American flying squirrels, who then infect humans. Diseases evolve as microbes find new vectors and, thus, hosts through the process of natural selection. For example, syphilis used to kill its victims quickly; after the disease evolved, victims began to die more slowly, allowing their spirochete to be transferred to more victims.