Brown fat contains an uncoupling protein that specifically allows protons that are pumped out to return into the matrix without going through ATP synthase. The reason you need protons to go through ATP synthase is because the protons actually mechanically power the "pump" action of ATP synthase. You might look into this, as there is a cool experiment where they attached a fluorescent reporter onto the end of ATP synthase and show that it actually rotates. There is a video in the supporting information of that particular paper that shows this beautifully.
With respect to the actual generation of heat, it can be best seen from a conservation of energy argument. Before, you had a proton motive force whose energy is translated into the mechanical energy of the ATP synthase pump. But now you have this thermogenin protein that uncouples the proton motive force from ATP synthase - it doesn't absorb the energy itself, so where does it go? It goes to the generation of heat, or dissipative energy. In other words, the proton motive force can be expressed the following way: proton(out) ---> proton(in) + energy. In ATP synthase, that energy is used to spin the lower part of the protein to force formed ATP out (rate-determining step in ATP synthesis). In thermogenin, that energy is released as heat. How? The normally exothermic reactions are now uncoupled from endothermic ones (ATP synthesis) and therefore release disordered heat into the surroundings.
Here is a very informative review if you are interested (beyond the scope of the MCAT):
http://physrev.physiology.org/content/84/1/277