How do the four stages of human diseases contribute to natural selection?

I think you're referring to the transfer of disease from animals to humans, a subject that Jared Diamond discusses in his 1997 Pulitzer Prize-winning book Guns, Germs, and Steel: The Fates of Human Societies.
To quote Diamond:

Given our proximity to the animals we love, we must be getting constantly bombarded by their microbes. Those invaders get winnowed by natural selection, and only a few of them succeed in establishing themselves as human diseases. A quick survey of current diseases lets us trace out four stages in the evolution of a specialized human disease from an animal precursor.

Let's review those four stages and how they contribute to natural selection. In the book, Diamond illustrates some of this information in "Table 11.1: Deadly Gifts from Our Animal Friends."
Stage 1: A human picks up a disease directly from a pets or another animal.
These include cat-scratch fever (from cats) and brucellosis (from cattle.) These are not spread from human to human, nor from animal to animal.
Stage 2: A former animal pathogen evolves to the point where it does get transmitted directly between people. Causes epidemics.
The epidemic dies when everyone becomes immune, or when everyone who's infected dies. The author uses laughing sickness (a fatal illness) in New Guinea as an example.
Stage 3: Former animal pathogens that established themselves in humans that have not yet died out.
May become "killers of humanity," to borrow Diamond's phrase. Lyme Disease is a good example.
Stage 4: Major, well-known epidemic diseases that are limited to humans.
These diseases are the winners, so to speak, in the process of evolution: they've survived when other epidemics and diseases have died out, as mentioned above.
Diamond uses the example of typhus, which was initially spread between rats via rat fleas. Those fleas could also infect humans. But typhus microbes "discovered" that human body lice was a better way to transmit the disease. And the microbes have continued adapting:

Now that Americans have mostly deloused themselves, typhus has discovered a new route into us: by infecting eastern North American flying squirrels and then transferring to people whose attics harbor flying squirrels.

This is natural selection at work. According to Diamond, "diseases represent evolution in progress," and "microbes adapt by natural selection to new hosts and vectors." Those microbes that figure out how to survive are the ones that are able to propagate. The typhus microbes that exist today are a product of natural selection, which is why it's still possible to hear about someone being diagnosed with typhus—and not, say, laughing sickness.