The full paragraph from J.K. Rowling is
Albus Dumbledore was never proud or vain; he could find something to value in anyone, however apparently insignificant or wretched, and I believe that his early losses endowed him with great humanity and sympathy. I shall miss his friendship more than I can say, but my loss is as nothing compared to the Wizarding world’s. That he was the most inspiring and the best loved of all Hogwarts headmasters cannot be in question. He died as he lived: working always for the greater good and, to his last hour, as willing to stretch out a hand to a small boy with dragon pox as he was on the day that I met him.
The phrase "working for the greater good" means "devoting his time for the benefit of society as a whole."
The phrase "in his last hour" means "right until his death."
The phrase "stretch out a hand" means to "offer help".
The disease "dragon pox" is something a wizard knows how to heal.
The whole sentence means that Albus Dumbledore was always helping people. As he approached his death nothing changed, and he was as ready to give help then, as he had always been.
One phrase that is sometimes used is
He helped others until the bitter end.