Why is Booker puzzled?

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.