Why does a balloon get larger when it is heated?

This mostly has to do with the energy of the molecules inside the balloon.

Anything that has mass has some energy just by existing. Usually, if you add some energy to that mass, you can get it to move; if we compared two objects of identical mass, but one of them is moving, we can safely say that the moving one has more energy. This is the basic explanation for the various states of matter; solids have less energy and aren't moving as much as gases, which are moving rapidly and have so much energy that they just bounce off of each other when they collide.

This "bouncing" is identical to the air pressure we feel at all times. This is also why you have to exert some force to blow up the balloon in the first place; you have to resist the force of all those air molecules ramming into the balloon at high speeds. However, once you've filled the balloon, it should stay the same size; this is because the average energy of the molecules inside the balloon "pushing" on it from the inside, is equal to the energy from the outside.

When you heat up the balloon, you add energy to the molecules inside. Since they can't really get any heavier, that energy goes into their motion instead, so they move faster. This means each collision is even more powerful than it was a a moment ago, when the molecules were cooler. You also haven't changed the energy of the molecules on the outside; they're still pushing with the same energy as a moment ago. So, the hot balloon molecules start to "win" - they get to push outwards until they reach a new equilibrium point; the point at which their extra energy is balanced by the other forces, such as the distance they have to travel or the greater number of molecules pushing on them from the outside.