Harry comes to, naked and entirely alone in a bright mist. He hears a noise of something struggling and wishes he had clothes. Robes appear and he puts them on. He begins to look around and sees that the thing that made the noise is some small naked child that looks flayed, shuddering under a bench. It scares Harry, but he approaches anyway. Dumbledore walks toward Harry and tells Harry that he can't help and then leads him away. They sit down. Dumbledore confirms that he's dead, but he says that he doesn't think Harry is. Harry says he should've died and didn't defend himself, but Dumbledore happily says that will help. At Dumbledore's urging, Harry deduces that Voldemort killed his own soul. Dumbledore assures Harry they can't help the flayed creature.
It's important to keep in mind that this entire conversation is a product of Harry's brain; Dumbledore is dead and isn't actually talking to Harry. Because of this, everything that Harry figures out or learns in this conversation is something that Harry already knew—he just needed someone like Dumbledore to spell it out for him. In this way, this conversation represents a passing of the torch of sorts, in which Harry discovers his own ability to act as a mentor and figure things out on his own.
Harry asks how he's alive if nobody died for him. Dumbledore waits for Harry to realize that Voldemort took Harry's blood, which will keep Harry alive because Voldemort is alive. Dumbledore explains that Harry was the unintended seventh Horcrux. He says that Voldemort doesn't take the time to learn about things he doesn't care about, like love, loyalty, house-elves, and children's stories. He says that taking Harry's blood and with it, Lily's sacrifice, keeps her sacrifice alive.
Readers familiar with the series might remember that Dumbledore seemed momentarily triumphant when he learned that Voldemort took Harry's blood—this is why; he suspected that Voldemort, who cares little for love, wouldn't understand the true implications of doing such a thing and because of this, inadvertently gave Harry a foothold in life.
Harry asks why his wand broke Voldemort's borrowed wand. Though Dumbledore says he's not sure, he guesses that Voldemort strengthened his bond to Harry by taking his blood, and when they dueled that night, Harry's willingness to die changed both his and Voldemort's wands. Harry's wand recognized Voldemort when Voldemort tried to kill Harry the night Harry moved. They both agree that, minutes ago, Voldemort failed to kill Harry with his wand, and Harry says it seems like they're in King's Cross station.
What happened when Voldemort sent the killing curse at Harry moments ago is that the part of his soul encased in Harry died; Harry essentially died for himself and left Voldemort's soul vulnerable. Harry's sacrifice, however, also means that he's saving countless others whom Voldemort would like to kill, as Harry didn't intend to save himself—he intended to save everyone else.
Annoyed, Harry asks about the Deathly Hallows. Dumbledore looks guilty, asks for Harry's forgiveness, and suggests that he, like Voldemort, wanted to be the master of death. He says that Grindelwald came to Godric's Hollow pursuing the Hallows, as Ignotus Peverell's grave is there. Harry asks if he's the last of the Peverells, and Dumbledore confirms this. He says he had the Elder Wand and the Cloak when James died. Dumbledore says that Harry should despise him and reminds him of what happened to Ariana. He says he loved his family, but he was selfish and brilliant and Grindelwald was the same. They wanted to hunt for Hallows, and the ensuing fight killed Ariana.
In this moment, Dumbledore lays out how seeking the dead and digging into immortality can distract a person from how they should be living, given that he blames Ariana's death on his obsession with the Hallows. With this, he encourages Harry to continue to look forward to those who are alive and those who are yet to come, and to dedicate himself to helping them rather than reaching back for the dead—including for him. Harry, in other words, must move on from Dumbledore.
Dumbledore says that, in the years after, Grindelwald acquired the Elder Wand and Dumbledore refused to be Minister of Magic—he fears he would've abused his power. He says that he put off dueling Grindelwald because he was afraid of figuring out the truth of who killed Ariana, but he finally got the wand. They discuss that Grindelwald tried to keep Voldemort from getting the wand, and then Harry asks if Dumbledore tried to use the Resurrection Stone. Dumbledore says he was unworthy to unite the Hallows: he took the Cloak out of selfish curiosity, not understanding that its power is that it can protect more than one person, and misunderstood how to properly use the stone. Harry, he says, is worthy, as he used them to enable his self-sacrifice. Harry's anger with Dumbledore disappears.
The idea that the Cloak is so powerful as a Hallow exactly because it can protect others is one that, in retrospect, carries through the entire series: Harry has spent a great deal of time with Ron and Hermione under the Cloak and, in addition to using it for mischief, they've all used it for a great deal of good. This, Dumbledore suggests, is what elevates it above the other Hallows, which exist more to better the person that possesses them than to improve the life of anyone else.
Harry asks why Dumbledore made the journey so difficult. Dumbledore admits that he didn't want Harry to seek the Hallows for the wrong reasons and says that he doesn't think Voldemort knew about the Hallows at all. He thought that Voldemort would go for the Elder Wand because Voldemort thought an unbeatable wand would solve the problem better than asking what Harry has that he doesn't. Harry confirms that Dumbledore wanted Snape to end up with the Elder Wand. After a moment, Harry asks if he has to go back. Dumbledore says he has a choice. Dumbledore says that if Harry goes back, he believes that he can defeat Voldemort. Harry sighs and asks Dumbledore if this is real or his imagination. Dumbledore says that it's in Harry's head, but that doesn't make it unreal.
Giving Harry the choice to return to life or not shows that it truly is choices that define a person. While Harry will certainly die a martyr if he chooses to stay dead, returning to life will make him even more of a hero and will allow him to go on to do even more good in the world. Dumbledore's assessment of why Voldemort is struggling with the Elder Wand stands as a final reminder that relying on power alone isn't enough to make a person successful. Instead, a person needs to dive into theory and knowledge, and doing this will allow a person to properly possess the wand.