I don't like the whole "drive through" the punch concept, I think it tends to make newer people push their punches, especially on the bag. "Snap" refers to what happens at the end of your punch, when your arm fully extends and you turn your punch over. The "Snap" or "Pop" you get into your punches dictates whether it will be a powerful or a weak punch. If you're stiff and don't get any snap into your punches, you won't have any power.
Realistically, you should strive to get good pop into all of your punches. By "drive through" your punches, perhaps your coach is telling you to fully rotate your hips and extend your punch.