Microwave heating is largely caused by the changing electric and magnetic fields (i.e. the "microwaves") which are emitted by your microwave oven affecting polar molecules. As the direction of the electric field changes over time, the polar molecules (often, of water) attempt to follow the field by changing their orientation inside the material to line up along the field lines in an energetically favorable configuration (namely, with the positive side pointing in the same direction as the field lines). As these molecules change direction rapidly (millions of times per second at least), they gain energy - which increases the temperature of the material. This process is called dielectric heating.
However, water is not the only polar molecule in the world. You can test for yourself that most plastics don't heat in a microwave while most glass and ceramic objects do. So, a microwave oven melting your plastic bowl has more to do with it over-heating your food than over-heating that food's container.
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