It turns out that when Voldemort rebuilt his body in Book 4, Harry Potter and the Goblet of Fire, he used some of Harry's blood, which means that the tables have turned – Harry's alive because Voldemort unintentionally tied him to life with that move.
Harry, Dumbledore explains, is the seventh Horcrux (an unintentional one), and Voldemort never even knew it. It's just one of many things he never understood – like love.
In taking Harry's blood, Voldemort also kept alive part of Lily's protection, and thus keeps Harry alive now.
Finally, Harry wants to know about his wand: why was it able to break Lucius Malfoy's wand when Voldemort used it against him?
Dumbledore's not absolutely sure, but his guess is something along these lines: when Harry faced off against Voldemort the first time, and their wands recognized each other as having the same core, Harry's wand (the winning one) took in some of Voldemort somehow.
When it recognized its enemy later, it was able to spit back some of Voldemort's own magic at him, which overpowered Lucius's ordinary wand.
Harry asks where they are, and Dumbledore, in turn, asks Harry where he thinks they are. He looks around and thinks it resembles an empty, cleaner King's Cross Station. Dumbledore bursts into laughter, and Harry is annoyed.
To shut him up, Harry finally asks about the Deathly Hallows.
It works. Dumbledore is deadly serious. Dumbledore apologizes for never telling Harry about his secret quest – it was a selfish one, and a foolish one.
He admits that, in his own way, he was doing what Voldemort was – seeking to avoid death. Only with Hallows, not Horcruxes.
Dumbledore tells Harry that the whole story of the three brothers is true – and that Harry is the descendant of the third Peverell brother, Ignotus, whose Invisibility Cloak has been passed down through many generations until it came to Harry.
Dumbledore admits his own selfishness in craving the Hallows – and in his treatment of his family, and his relationship with Grindelwald long before.
Finally, everything is out in the open between Dumbledore and Harry. Dumbledore tells Harry about Ariana's death and its aftermath, and his subsequent break from Grindelwald. He's ashamed, deeply ashamed.
Harry, trying to make him feel better, tells Dumbledore what he saw in Voldemort's vision – Grindelwald's final resistance to the Dark Lord.
Dumbledore explains the moment when he finally possessed all the Hallows. He realizes that he wasn't fit to be the Master of Death – he could only master the Elder Wand, but not the other two Hallows.
Only Harry is the true master of all three Hallows, because the real master of Death doesn't run away from it.
The only part of Dumbledore's plan that didn't work out was the Elder Wand. He'd wanted Snape to end up with it, not Voldemort.
Finally, Dumbledore tells Harry that he has a choice – he can either go back or stay.
Harry knows he must go back; Voldemort has the Elder Wand, and everything possible must be done to defeat him.
Right before he leaves, he ask Dumbledore if this is real or just in his head – the old Headmaster answers mischievously that it is both.