Liquid nitrogen can't freeze fire, but it can put out a fire. The reason liquid nitrogen can put out the fire may be surprising; It's not because it's cold, but rather because it smothers the fire, preventing the fire from getting enough oxygen to continue burning.

You can't freeze fire because fire is made mostly of heat and light, which come from tiny glowing hot particles. You can't freeze heat or light because they are energy and not matter.

The particles are matter, but they are usually so small that you can't see them with your eyes. You only see the light that they give off. This is the case for a clean-burning fire, like the one on a gas stove-top.

It's possible that for a VERY sooty fire, with large burning particles, you'll see some particles fall after pouring the liquid nitrogen on it. They fall because they cool off and no longer get lifted up in the hot air of the fire. Sometimes this happens naturally after a volcano or forest fire. Big particles of wood burn and rise during the event, only to settle down again after cooling to create a blanket of soot.