I am not too sure about this, since it has been a while since I had my combustion course but here is my answer.
Every chemical reaction creates 3 major products: Nitrogen, Water, and Carbon Dioxide. My assumption (and I think some professor had actually mention this before) is that when you shut off your car, some of the combustion residue do not flow out of the exhaust system completely and is left somewhere in the system, including water. When you first start up the car in a cold weather, all the parts that the water molecules are attached to are cold, and therefore when the hot exhaust gas pass through the exhaust system, you boil the water and create steam, hence the water steam you see out of the exhaust pipe when you first start the car. Once all the components are up to operating temperature, there are no longer water condensing on the wall of the exhaust system, and therefore, no water to boil and no visible steam (water is now include in all the by product).