How did Pasteur and Koch prove that a specific germ causes a specific disease?

With two kinds of proof: Pasteur showed that microbes do not appear from nothing, and Koch showed that one particular microbe can be isolated, grown and shown to cause one particular disease. Before that, spontaneous generation, the theory that living creatures could arise from non-living matter, was commonplace. Pasteur's flask experiments demonstrated that in sterilized and sealed flasks nothing ever developed, while in sterilized but open flasks microorganisms could grow. Germs, in other words, come from other germs that arrive from outside.

That left the second question: does a specific germ cause a specific disease? Robert Koch studied anthrax, which near Wollstein was regularly taking the lives of humans and livestock without evidence explaining why. In 1876 he discovered that anthrax was triggered by one singular pathogen. To study such organisms he developed a way of growing bacteria on agar and glass plates, which his assistant Julius Richard Petri later turned into the Petri dish.

He then turned those methods into a recipe, known as Koch's postulates: four criteria designed to establish a causal relationship between a microbe and a disease. The microbe must be found in the sick and not the healthy. It must be isolated and grown in pure culture. The cultured microbe should cause the disease when introduced into a healthy host. And it must be re-isolated from that host and identified as the same agent.

They have limits: Koch himself abandoned the first requirement when he found asymptomatic carriers of cholera, and viruses cannot be grown alone in pure culture.