Initially you have to heat whatever you are burning to a gas that is at or above it’s flash point. This process generates a lot of combustible vapor that isn’t quite at the flash point yet. Once the fire starts and flares (burns up the excess vapor) it will die back to the point where it consumes the vapor as it is generated while not allowing an excess to build up like it did before the flare.
Basically, though, I'm concluding that even though fire may spread, it is really dying -- fortunately. Nascentes morimur, so to speak: "from the moment we're born we begin to die."