At this tiny scale, energy and mass are essentially equivalent. So when we say that matter annihilates, we mean that they transform into pure energy (in this case, as photons of light). They don't break into subatomic particles, because that still counts as mass. They just simply cease to exist.

As a side note, the "conversion rate" of mass into energy (and vice versa) is governed by Einstein's E=mc^2. All this equation means is that it takes a ridiculous amount of energy to create a small amount of mass, and vice versa, it only takes a small amount of mass to create a ridiculous amount of energy. Because antimatter annihilates completely (ie, 100% of its mass, as well as 100% of the regular matter's mass, gets converted into energy), antimatter is currently the most explosive thing known to mankind

If I understand it correctly, annihilation is a 100% efficient process that converts all the matter into energy. After the process is complete there is no matter left over and only energy in the form of light, heat, and other energy forms that go way over my head remains.

Annihilation means exactly that - both particles destroy each other on contact, releasing the energy that composed them.

For the simple case of electron-positron annihilation, they transform into high-energy photons, whose total energy is equal to the total mass-energy of the electron and positron. See: https://en.wikipedia.org/wiki/Electron%E2%80%93positron_annihilation

While atoms can be comprised of antimatter the interactions are generally on a subatomic level, i.e. electron/positron, and proton/antiproton. Since particles/antiparticles are identical to their counterparts aside from charge any such interactions are total with nothing left over other than the resulting energy release usually in the form of photons. The results of an atom reacting with an anti-atom could have a variety of results depending on the differences in weight between the two. Exactly what those results might be is a bit beyond my lay-understanding of the process.