Air activated hand warmers work by reacting iron with air to form iron oxide, which you may know as rust. The reaction is exothermic which means that it releases heat.

You may be wondering why these hand warmers work even though under normal conditions iron does not rust simply when exposed to air.Usually it takes water, air, and iron to form rust and even then the rust forms slowly. The answer is that the hand warmers contain water so that they can form rust and also a catalyst which speeds up the reaction so it happens fast enough to keep you warm.

Here is a website with a more detailed explanation:

Air-activated hand warmers contain water and elemental iron. The extremely fine particulates of iron, particularly with salt as a catalyst, cause a rapid oxidation of iron to rust from oxygen in the atmosphere, a chemical reaction that generates heat.