Most teeth whiteners work with peroxide molecules (like hydrogen peroxide). Peroxides are very oxidizing. Most "colored" molecules (stains) are oxidizable, such that their oxidation product is colorless. When the stain molecules come in contact with peroxide, they become oxidized, making them colorless.

Along with this also comes the risk that the peroxides will oxidize other molecules that you need to function normally, so there is create care that is taken to make sure that the teeth whiteners aren't too effective/reactive.