How come people use air to blow out small fires, but also use it to make large fires bigger?

I have noticed that, too. And at first it does seem to be a surprise. Air does two opposite things to a fire. It provides oxygen that is needed for anything to burn. If you keep air from getting to a fire, you can put it out.

Air also can cool down a fire. The flame and heat of the fire come from burning gases. A blast of air hard enough to keep the flame away from the fuel will put out a fire.

I guess this is another case of “a little is great, but too much is bad.”