Food digested in the dragon’s stomach is turned into gases. The (flammable) gases are stored in a organ that is connected to the throat. When the dragon wants to breathe fire, it releases some of the gases, which go through the tube connected the throat. When it hits a point near he top of the throat, it ignites and comes out as flame. The dragon breathes out at the sme time to speed up the flame and send it farther from it's body.

Another Description of how a dragon breathes fire.