How did fires go out,...?

Question

back when there was no people?

Answer 1

Many fires probably burned until the season changed.  Others would have followed their fuel supply until it reached a pinching point.  A change in wind direction or lots of rain could have also put them out.

In any event, wild fires would have covered much larger areas than they do today.  Which wasn't necessarily bad...range and forest fires are actually very important in some ecosystems.

Answer 2

They either ran out of fuel....or heavy rains.