Most chemists now agree that microwaves are not doing anything mysterious to chemicals. They are simply another form of electromagnetic radiation, like light, infra-red or radio waves.
Microwaves have a wavelength between 1 mm and 0.1 m (corresponding to a frequency of 300 to 3 GHz). At these wavelengths the electromagnetic energy interacts with polar molecules, such as water, making them spin. The spinning water molecules knock into each other and pass on their rotational energy by making their neighbours vibrate. The more molecules vibrate the hotter they get.
But, while there is nothing mystical about this, the process of microwave heating means that the energy is transferred to the molecules more effectively than simply heating them in a flask with a Bunsen burner so that the reactions generally go faster.
The formation of hot spots in a reaction mixture means that there are also very sharp rises in temperature without the solvent having a chance to boil away. Water, for instance, can reach 105 Celsius before boiling in such a hot spot in the microwave. The common organic solvent acetonitrile can reach 38 degrees above its normal boiling point.
Since most reaction rates are accelerated by increasing the reaction temperature – a ten-degree rise in temperature, doubles the reaction rate – a hot spot at a temperature above the normal boiling point of the reaction solvent means a microwave reaction can be that much faster.