There are two physical processes that are important in a microwave oven. First is dielectric heating: basically, the oven sets up an electromagnetic wave inside itself, and certain molecules in the food absorb energy from the wave. Specifically, an absorber molecule needs a nonzero electric dipole moment, and it has to be free to rotate. Liquid water is a very efficient microwave absorber, fats and oils are somewhat less so, but a lot of molecules are not. So only some of the molecules in the food can actually absorb energy from the microwave oven.

The other important process is called thermalization. All that means is that the hotter molecules (which in this case happen to be the absorbers) will move around and bump into other molecules, and transfer some of their energy away. This is how heat spreads from the absorbers throughout the food.

Now, the thermalization process takes some time, typically more time than the food actually spends in the microwave oven. So when you take it out, it's unevenly heated: some of the molecules still have more energy than others. The temperature does even out after some time, but that means that the energy from the absorbers has to be shared out among all the molecules in the food (and its container), so they're not as hot as they were when they came out of the oven. This is the cooling process you've noticed.

In contrast, when you heat food on a stove or in a conventional oven, the only process involved is thermalization, and every molecule in the food (and pot) is equally capable of receiving energy by that process. So when you take your food off the stove, all the molecules have been completely heated up. The energy doesn't have to be "shared out" any more. For that reason, the food has more total thermal energy which keeps it hot longer.