This quote shows up near the end of part 4. The plague or pandemic in this instance is, on the surface, an outbreak of the bubonic plague. But the plague also symbolizes the spread of Nazism, showing the way an authoritarian politic ideology can be transmitted across a society with lethal results.
This quote is part of a long speech by Tarrou, who wants Rieux to know who he is as a person, what drives and motivates him. He tells the story of the identification he felt with a man found guilty of a crime when he, Tarrou, went to watch his father in court. In a sudden leap of empathy, Tarrou was able to identify with the "defendant" as a living, breathing human being just like him. At that moment, he was repelled at his father's authoritarianism: his father became a man in his official, professional capacity who wanted nothing more than to kill another human being by imposing the death penalty.
The plague that Tarrou perceives, the germs of which he says are latent in everybody, including himself, is the desire to impose cruelty and death on other humans, which is made possible by reducing other humans to dehumanizing labels such as a "defendant." Tarrou says that "no one can ever be immune" from these cruel desires. He goes on to state that:
we must keep endless watch on ourselves lest in a careless moment we breathe in somebody's face and fasten the infection on him. What's natural is the microbe. All the rest, health, integrity, purity (if you like), is a product of the human will, of a vigilance that must never falter.
In other words, this evil of wanting to harm others is natural to us, the human condition, and a part of all us we must keep a constant watch over so it doesn't take over a society.