Tonsil stones, or tonsilloliths, are formed when food gets squeezed into cracks or holes in the tonsils during the second phase of swallowing, where quite high pressure form in the back of the mouth, right next to the tonsils. This food, being organic, decays with time, and starts to taste and smell bad. The decaying process can also set up an inflammatory reaction and trigger a sore throat or tonsillitis.
This picture shows a crack in the right tonsil. If you zoom in you'll see small greyish rounded structures within the crack. These are tonsil stones.