He'd been fighting the Time War for centuries and his line "wearing a bit thin" implies (like his first incarnation) that he was simply dying of old age. His body had been ravaged by the effects of the war and so likely his old body just gave out.
In addition to the length of time that had passed in his timeline between "The Night of the Doctor", when he regenerated into the War Doctor, and "The Day of the Doctor", he had been exposed to a Time Destructor in "The Innocent" (audio story). This weapon has the effect of causing very rapid ageing (as shown in "The Daleks' Master Plan"). Although the ageing is usually fatal, a Time Lord can survive it if the exposure isn't too great. Even a Time Lord will suffer harm from it, however, and the War Doctor had to spend 50 days in an induced coma to recover.