Dumbledore leads Harry to a couple of seats and commends him for his bravery. He acknowledges that he is dead, but says that Harry is probably not. He explains, or helps Harry to figure out, that while Voldemort has just killed the part of his own soul that was embedded within Harry, Harry is still alive because Voldemort reconstituted his own body out of Harry’s blood (in Harry Potter and the Goblet of Fire). Because Voldemort contains Harry’s blood, as long as Voldemort is alive he preserves Lily Potter’s charm, so Harry can’t die at his hand. Thus, paradoxically, while Harry had to die before Voldemort could, Harry can’t die while Voldemort lives. They have a double bond, with Voldemort’s soul in Harry and Harry’s blood in Voldemort.
Dumbledore explains the mystery of why Harry’s wand defeated Voldemort even when the latter had Lucius Malfoy’s wand. The first time Harry fought Voldemort with their twin wands, Harry won because his courage was greater. Because of Harry’s bond with Voldemort, and because of the kinship between their two wands, Harry’s wand absorbed a bit of Voldemort’s essence and also came to recognize him as a mortal enemy. That is why Harry’s wand recognized Voldemort and defended against him, turning a bit of Voldemort’s highly potent magic back against him and destroying Lucius’s wand.
Harry raises the subject of the Deathly Hallows, and Dumbledore admits with shame that he was seduced by their promise to make him master over death. The search for the Hallows drew Dumbledore and Grindelwald together years before, and they had intended to embark on a search for them when Aberforth pointed out that they couldn’t leave Ariana. Dumbledore, realizing that the craving for power was his most dangerous weakness, turned down the post of Minister of Magic and stayed at Hogwarts his whole career.
Dumbledore avoided facing Grindelwald for as long as possible, afraid that he might learn that it was he, Dumbledore, who cast the spell that killed Ariana. Finally, he defeated Grindelwald and took the Elder Wand from him. Dumbledore had given up on the search for the Hallows when he learned that Harry’s father had the Cloak and borrowed it to examine it. When Dumbledore got hold of the ring with the Stone, he couldn’t resist using it to try to speak to his sister and parents. He put it on, forgetting that the ring was now a Horcrux and thus cursed, thereby ruining his hand and causing his own eventual demise. He says that he never could have united the Hallows because he took the Cloak out of idle curiosity and the Stone for selfish reasons, wishing to disturb the peaceful dead. He only did the right thing with the Wand, having taken it to protect others from it. Harry, on the other hand, only wanted each of these items for selfless reasons.
Dumbledore concludes by explaining that he had counted on Hermione to slow Harry down somewhat during his quest, keeping him from rushing after the Hallows, so that Harry would not impulsively seize upon the Hallows for the wrong reasons. He says that Voldemort just wanted a wand powerful enough to beat Harry, while understanding nothing of the Hallows. Dumbledore admits that he hoped that by having Snape kill him, he could protect the Wand from being taken by another unscrupulous master, but that things hadn’t worked out as he’d planned.
Finally, Dumbledore tells Harry that he can choose to go back to life or move on. In answer to Harry’s question, he acknowledges that all of this is happening inside Harry’s head, but that this fact does not make the conversation less real.