Alright, these have most likely been answered, but i'll post for the heck of it:
-Why did Booker have to die at the end?
The realization comes to Booker and Elizabeth that the only way to get rid of Comstock is to prevent him from coming into existence. Booker says he'd go so far as to smother him in his crib. When the revelation hits that in some timelines/universes (whatever term you want to use), Booker becomes Comstock, killing him (before that moment of truth where he heads down the path to becoming Comstock) erases Comstock from...existence? I guess. But a true multiverse theory suggests that there'll always be realities out there where this act didn't happen, and an infinite number of Comstocks and realities with Comstocks still exist and always will.
-How could he turn into Comstock?
Many have postulated that baptismal rites and 'being reborn' after traumatic events (Booker's participation in war) could seriously turn a man nuts.
-What is "Anna"? Is that Elizabeth's real name or something? How could she be his daughter?
She is his daughter, one that he (the protagonist, Booker) sold to pay off a gambling debt. In the ending, you see that he tried to change his mind as he relives the moment, and tries to get her back from Lutece and Comstock. Being pushed around between realities has fucked Booker's mind; his nose bleeds for the last half of the game, like the soldiers you've killed who are dealing with multiple realities/outcomes in their brain. The first quote in the game from Lutece (and one printed on the inside of the game's box sleeve) suggests Lutece's theory of the mind manufacturing new memories to deal with missing ones, or something like that. We can theorize that Booker created new memories to replace the gaps and trauma created by losing Anna, I suppose, which is why he forgets about that all for the majority of the game.
-What was the point of killing song bird? Just to give Rapture a cameo?
I...suppose. I mean, it ends Liz' story well; she's free. Destroying the tower itself was crucial. The siphon was destroyed with it, releasing all the energy drained from her back to her body, allowing her to become omniscient again.
-who/what are the British people that show up throughout the game? Can't remember the names.
Oh boy. These are the Luteces. They're kind of super-pivotal to everything that happened. The suggestion is that Robert is the parallel version of Rosalind from another reality, one where his Y chromosome was present and thus, was born a man.
Rosalind was the scientist who met Comstock, and developed a) Columbia/floating cities, and b) Tears/Quantum Physics. She was the catalyst for the whole game's events, basically.