Brain's reaction to hand transplant

That's assuming an either/or where there isn't one. Brain regions don't have one function. If they did it would make our jobs that much easier! Instead, the amputation likely dampened activation associated with the hand (phantom pain results when it doesn't) and begins to respond to facial stimulation. But there's still some input to that region from the rest of the arm. By plugging in a new hand, the old region begins receiving very similar, but more extensive, input since the connecting nerves, spinal cord to brain, are intact. It's just a matter of the volume and frequency getting turned up. The more difficult part is the brain re-learning the fine motor skills because:

Savage’s recovery so far is limited to major nerves in the right hand, not to peripheral nerve connections for individual fingers, Frey says. It’s unclear how the neural map of the right hand will adapt as Savage’s finger nerves regenerate and finger sensation develops.