"Driving off the front foot" is a simpler way of illustrating how a QB generates power in a throw. Arm strength is about 30% the strength of the throwing arm, and the rest mechanics.
The first mechanism that power is generated through is through the torquing of the hips. In college our QB specific lifting program included many exercises and lifts focused on hip explosiveness, which was great on the field (and had benefits off the field as well). But the very first movement a QB makes when starting a throw is opening up the front hip. The more compact and explosive that movement is, the more arm speed can be generated.
Next is the lead arm. This is often neglected, but is essential to a throw. As, say, a right-handed QB makes a throw, he can use that left arm to "pull through" the throw. This also generates power. As the left arm pulls the body forward, this also generates torque/power.
Driving through the front foot is simply the completion of the movement. If a QB can really step into the throw, it is a four point process: hip turn, lead arm pull, throw, and follow-through including the back foot finishing through and in front of the front foot, and the QB generating the last bit of power when it is the front foot, and not the back foot, pushing off the ground. Think about it, at the moment of release it is the front foot solidly on the ground, not the back foot. Mechanically sound QBs are still generating power at that moment by pushing forward off the balls of their front foot.