Let me offer a simplified explanation, if I may. There are two functions that are cooling the ocean following a tropical cyclone. The first, and obvious one, is mixing at the surface. Waves, wind, turbulence, etc cause the mixed layer (an isothermal and isosaline layer) to deepen into the lower layers of the ocean. This mixing introduces water that is usually not affected by the sun and cooler, thus bringing down the temperatures of the mixed layer.
The second, and more significant effect for a strong storm, is Eckman Pumping. This is a result of
Eckman Transport, which is a forcing of the surface water by wind. Basically, in the Northern Hemisphere, water is forced at a vector ~45 degrees to the right of the wind. If you picture a Tropical Cyclone in the Northern Hemisphere, the winds are circling counter-clockwise (cyclonically) and the water is being pushed to the right, or away, from the storm. If all the water is being pushed away from the center of a storm at the surface, the only way new water can come replace it is from below, and the water from below is colder. Much colder, in fact, than the water from surface mixing. Hence the cold wake.