I use my microwave every day to heat up coffee, leftovers, stuff like that, and I've always found it interesting how it actually works. The story's way simpler than it seems: a microwave generates electromagnetic waves at a frequency of around 2.4 GHz. These waves don't "burn" the food like direct heat - actually, they make water molecules (and a bit of fat and sugar molecules too) vibrate really fast. When a water molecule vibrates like that, the friction between them generates heat. It's like rubbing your hands together to warm up in the cold, but happening billions of times per second inside the food.
What makes it fast is that the heating doesn't come from the outside in, like in a regular oven. The waves penetrate a few centimeters into the food and excite the water molecules right there inside. That's why a plate heats up too (it has moisture), but dry plastic doesn't heat up as much - there's no water molecules to vibrate. And then there's the myth about it being dangerous? Well, microwaves are non-ionizing radiation, totally different from X-rays or nuclear radiation. They don't transform atoms into other things. As long as your oven has the metal mesh protection on the door (which blocks the waves), you're getting zero radiation.