At the end of "The Day of The Doctor", we see John Hurt's War Doctor step into his TARDIS and start to regenerate into Christopher Eccleston's 9th Doctor. I always thought that Time Lords can only regenerate when they are close to their death (old age, severe physical injury, etc.,) which isn't what's happening. Other Time Lords can also enforce a regeneration, but that's clearly not the case here since they are "gone." Why then did this regeneration begin?
It's a reference to the First Doctor's regeneration.
The First Doctor's explanation for why he had to regenerate was that "this old body of mine is wearing a bit thin." It's the end of a harrowing adventure (his first encounter with the Cybermen), and the man's had a really rough time of it all around. The idea is supported because Hartnell's failing health was becoming increasingly obvious to those watching the show. Hurt isn't as feeble as Hartnell was, not by a long shot, but he plays the part of an equally ancient, tired man who has seen many lifetimes' worth of struggle in that body.
But in the end, it's a callback line more than it is a good explanation.
The real reason is that he's the Doctor again.
As we saw in Night of the Doctor, the War Doctor does not consider himself the Doctor and never has. He specifically regenerated into that form so that he could do un-Doctor-y things. The events of Day of the Doctor restored his ability to be the Doctor, and he couldn't do that in the War Doctor's body (which was tailor-made to be a warrior by the Sisterhood of Karn).
Regeneration is not only available at time of death.
In fact, the only other Time Lord we saw regenerate on screen in Old Who (Romana I to Romana II, at the beginning of Destiny of the Daleks) seems to have done it on a whim. Regeneration can be triggered "manually," and it seems that many Time Lords treated it pretty casually. The Doctor's dangerous life of adventure (one which sets him apart from nearly every other Time Lord) leads him to use it more sparingly, keeping regeneration as an emergency get-out-of-death-free card.
It's his reward isn't it?
At the beginning of the story the Doctor Hurt tells Rose/Bad Wolf/Moment -
"I have no desire to survive this" and she/it replies "Then that's your punishment, if you do this, if you kill them all, then that's the consequence: You live."
Given that this specifically refers to the destruction of Gallifrey ("... killing them all, Daleks and Time Lords alike" as she/it says) and he doesn't do it, then logically he shouldn't be punished, so should die rather than live.
Thus he doesn't have to continue being the Doctor who fought in the Time War, but can be a new man who can put some distance between himself and the other things he did in that War ("I've been fighting this War a long time; I've lost the right to be the Doctor," he says.)
Of course, it's a bit double-edged, because the Eccleston Doctor is a bi-polar PTSD sufferer who thinks he's committed double genocide (due to not being able to "remember that I tried to save Gallifrey rather than burn it") and it takes a second Bad Wolf induced regeneration (in The Parting of the Ways) to allow him to get over that.
Outwith the fiction, I suspect it comes from a combination of the fact that we know this story was conceived (Moffat has said as much in DWM) with Eccleston's Doctor in the Doctor Hurt role (and thus there was no regeneration at the end originally) and a desire from the production team to 'cap off' the War Doctor with a regeneration scene. After all, Moffat has spoke of his own fannish desire to see McGann regenerate, surely the same impulse applies? It means that what happens is thematic rather than strictly plot-based, but it does work, I think. Albeit it is very surprising the first time you see it. (Doctor Hurt's line "Of course. I suppose it makes sense. 'Wearing a bit thin'!" feels more like placeholder than explanation.
It also leaves the rather odd disconnect in the story that Tennant and Smith do not realise that Doctor Hurt is from before the end of the Time War, even though we later see he dies immediately it ends. They think he could be from after it. That makes sense if he's Eccleston. Less so if it's Hurt.
There is no clear indication as to why he regenerated at the end of the episode... All he said was something along the lines as "I suppose this was wearing a little thin"...
Perhaps the answer lies in the question why did the War Doctor age so much from the prequel to the Day of the Doctor...
There is some speculation that he had aged a lot during The Time War or that perhaps he was hit with some sort of ageing weapon by The Daleks..
Because the time war was over. There would be no need for a warrior anymore, but a return to the Doctor. It has been shown several times old series and new that time lords can use their regeneration energy at will.
I think it was to do with the fact that he was old, but it might be also that he actually wanted to regenerate, because he accepted himself as the Doctor, and wanted a fresh start.
The War Doctor did not exist as a normal regeneration, but as one specific to the elixir given to Eight in 'Night of the Doctor'. He existed to fight the Time War. With it ended, his time was over. It is possible that is why he had more of a demeanor of acceptance for this regeneration than many of the others.
He thought he had doomed Gallifrey
The War Doctor was forgetting what really happened during his interaction with his later incarnations, as his true memories were being replaced by the false memories of pushing the button that had ended the war and doomed his world. He was dying from the stress of guilt and grief.
The War Doctor's comment about "wearing a bit thin" may have been a reference to his true memories. In the next scene, the Tenth Doctor remarked that "I won't remember either".