At the time, all events in the world were only possible because the Tardis was being used to power a paradox machine, allowing past and future to coexist and destroy each other without affecting the natural progression of things.
Both the Doctor and the Tardis have telepathic abilities, and everyone on the planet was linked by a group of satellites that connected them in a telepathic field. Everyone concentrated on the stories and said that word, envisioning in their heads what the stories meant to them. Those feelings raged across the planet as a primal force, enabled only by the satellite network and so many minds thinking the exact same thing - a saviour, the Doctor rather than their previous controlled state of fear.
The Tardis and the Doctor are linked so, when that energy entered him it entered the Tardis in a way too. The paradox engine combined with his knowledge of fluid time and a physiology built around regeneration, allowed him to use that energy to undo the unnatural passage of time on his body as if it had never happened (much the way that the year ended up not ever having happened). The psychic energy of the people doesn't need to know what he looks like to fix him, he only had to have the time to access the network and imprint his own self-image upon it.