The phantom limb is a continuing sensation of an amputated body part. These sensations may only occur if the relevant portion of the somatosensory cortex recognizes and is responsive to alternative inputs. The mirror therapy can apparently used to relieve the pain that people experience because it tricks their brain into thinking that the amputated leg or arm is actually the one doing the movements as well. I would think it would be because of the current sensations are binding with the memories of previous experiences. The synapse that used to receive input from the new amputated part become vacant is what makes the phantom limb sensation. Axons representing another part of the body take over those synapses.
I happened to find a video that demonstrated this kind of therapy that is being done on soldier who has had their leg amputated.