How did The Doctor revert his aging?

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.

I don't think you're ever going to get a clear answer to this one but I'll do my best.

The Doctor has already proven he has telepathic abilities (The Girl in the Fireplace). The Master has linked together the entire human race using the Archangel network, this is designed to subdue them.

During his year as a captive The Doctor integrates with this network so he can use it's psychic energy. When the countdown ends the entire human race thinks of The Doctor effectively supplying him with energy. With this energy his own natural abilities are boosted.

1. He can reverse his age (this could be considered a type of regeneration - after all in Let's Kill Hitler River Song mentions she's going to throw in a little reverse aging!)
2. He can protect himself from the Laser Screwdriver
3. He can float onto the steps

He also doesn't have to physically do anything to defeat The Master, who recognises that The Doctor not himself now control the population. Instead of fear of The Master they're full of hope for The Doctor.

To answer the direct question - I'd say he uses the energy of the network to reduce his age in the same way River Song will. This is most likely a natural timelord ability which he is able to use to the extreme because of the energy he has access to. The other parts are harder to explain and I'd write them off as fiction.

I find this explanation on Wikia fairly acceptable:

But as the clock counts down, Martha starts to laugh out loud, prompting the Master to angrily demand to know what she finds so funny. Martha now reveals the real reason she travelled the globe: it wasn't for a fictional anti-regeneration gun, with the Doctor himself pointing out that he would have never asked Martha to kill, nor was it to fight back; but merely to talk. She told everyone about the Doctor; specifically, she told everyone to think of the Doctor at the same time the Master plans to launch his fleet.

Docherty's betrayal was planned — engineered by Martha so that she would be brought on board the Valiant to rejoin the Doctor. Combined with the Master's Archangel Network, which the Doctor has had an entire year to tune to the frequency of, this has the effect of charging the Doctor with the combined psychic energy of the people of Earth. This enables the Doctor to restore his youthful physiognomy and end the Master's control. The combined psychic energy gives the Doctor great power and he is not only able to restore himself with it, but it shields him and temporarily grants him telekinesis with which he removes the Master's laser screwdriver from his hand.

As the Master cowers against a wall, the Doctor says the words the Master was terrified to hear: "I forgive you".

Source: http://tardis.wikia.com/wiki/Last_of_the_Time_Lords_(TV_story)