Thermogenin acts to short circuit the normal production of ATP via the ATP synthase. It, instead, allows protons to fall down their gradient, releasing heat instead of making ATP.
Work Step by Step
Consider two facts: 1) protons move, but not through the ATP synthase, 2) heat must be the end product. Protons move passively, which, from earlier material, means down a concentration gradient. And, any process (chemical, mechanical, etc.) produces heat even if no real work is done (think of how an idling car engine gets hot even though it's not moving the car). So, the uncoupling of the proton gradient being run down from the production of ATP must mean that the fat is broken down to produce heat via the action of Thermogenin.