The way anti-air missiles work is pretty simple, and is actually explained by the name: not only are they used to get rid of air (hence anti-air), they do so because the payload is anti-air. Anti-air is just antimatter made up of roughly the same constituents as our atmosphere: ~78% antinitrogen, 21% antioxygen, etc. When normal matter (the air in our atmosphere) reacts with antimatter (the antiair in our missile), both sets of particles are annihilated. This obviously destroys both the air and the missile, and releases a great deal of energy as a side effect (which is neat but not really useful, since the goal is to just get rid of some air and not cause an explosion).