So I understood what was happening in the game and the ending, my only question is:
Why didn't he remember a thing when he was brought to Columbia? We are playing the Booker who gave away his daughter to Comstock, as far as I know, so how can he misinterpret the "pay the debt" line? Why can't he remember that he sold his daughter? Why did he forget about the Lutece twins?
The jump between realities gives people a type of amnesia the first time they go through.
Refer to the dialog in this scene:
Booker: and wipe away the debt...bring us the girl, and wipe away the debt...
Robert Lutece: See? He's starting to put his story together.
Rosalind Lutece: Hm. You're quite fond of this theory of yours.
Robert Lutece: He's manufacturing new memories from his old ones.
Rosalind Lutece: Well...the brain adapts.
Robert Lutece: I should know. I lived it.
The Lutece 'brother' experiences it the first time he goes through, and so does Booker. In both cases, rather than realizing they have amnesia, their mind simply fills the gaps by creating new, false memories in their place as a coping mechanism.
In Booker's case, when the twins pull him into Columbia and his memory is affected, his mind makes the connection with the Lutece brother (and the fact that he gave the brother Anna to wipe his debt), replaces Anna with Elizabeth in his mind, forgets who the request came from (attributing it to a shadowy figure), and simply goes along for the boat ride, where he's dropped off to go hunt down Elizabeth.
After twenty years has passed since losing Anna, the Lutece twins open a portal in Booker's office and offered him redemption to reunite with Anna. The jump scrambled his brain, confusing him to believe the deal with Comstock 20 years ago was a rescue/kidnapping job in his present time (1912).
"The mind of the subject will desperately struggle to create memories where none exist..."
Adding on to lionel's answer,
The world in which Booker goes out to Columbia is the "get the girl, wipe the debt" Booker. He still thinks that he's going to retreive a girl from Columbia, to pay back his debts (most likely gambling/alcoholism). Another universe exists where Booker instead gave Anna to Comstock , then tries to get her back, but that's an altogether completely different universe.
The reason for those is hinted at the first quote you'll see when starting the game.
Booker was transported through a tear to another universe, and his mind had to make up memories to deal with traveling to another universe.
Notice that Booker did not know what or where Columbia is (he had to ask the first man he saw in Columbia where he is), and has shown surprise on seeing a flying city. This is despite Columbia being a significant part of the United States (before its secession) and took part in a major battle of that period, the Boxer Rebellion. It is probably safe to say that Columbia did not exist in the universe Booker was from.
To make sense of this, Booker had to modify or create new memories to deal with being transported to another universe. Another answer has already mentioned that Robert Lutece (who was also transported to another universe before Booker has), has experienced this.
Another example of Booker making up new memories is when he was transported to the universe where he is a martyr of the Vox Populi. He then "remembered" that he is/was a hero in that universe and fought with Slate (instead of fighting against Slate in the previous universe).
In the end of the game, Booker finally completely "remembers" what really happened, when he was transferred back to the day of his baptism. This event was a turning point for Comstock/Booker. One took the baptism, and the other didn't. More details regarding the ending and the timelines as posted in my answer on another question about the game's ending here.
