Just by observation.

Even without germ theory per se, back as far as ancient China, Mesopotamia, and Egypt, people could tell that exposure to SOMETHING caused disease. Night air? The moon? Miasma? Insect bites?

Keeping the sick away from everybody else, and noting that people got sick around sick people doesn't require the direct knowledge that micro-organisms are responsible.

Trial and error

They discovered that people who kept clean didn't get as sick

The Pope famously hid in a circle of flame to keep him safe during the worst of the plague. The theory was to keep 'bad air' away, but in fact it was keeping him safe from the rats