Here's my interpretation, the best I can do with this great, bittersweet game (apologies if this has been stated before, but long thread is long):
The Luteces have been trying a long time, with multiple Bookers from multiple Universes where he gave up his daughter, to save Elizabeth and bring down Comstock. All of those Bookers have failed until us. We are the Booker who succeeds. We are the Booker who makes it past dying for the Vox, past getting killed by Songbird, and makes all the correct moves to close the loop.
Each choice that we/Booker makes creates divergent paths that created many other Bookers. But we are experiencing the Booker that made all the right choices and went down the necessary path. He makes it to the end, sacrifices himself to close the loop on all the Bookers who went down to the river to get baptised, thus ending the cycle.
Our Elizabeth still lives because if she was erased from existence, then our Booker doesn't exist, and none of the game events happen and a version of the grandfather paradox ensues (if i go back in time to kill my grandfather, then I wouldn't exist to go back in time to kill my grandfather, therefore I exist to go back in time to kill my grandfather ad infinitum). This seems to be the popular interpretation around here because people seem to think that the cycle just continues when you hit New Game. But remember, this isn't one timeline, this is multiple Universes. Multiple Universes where memories bleed over and our Elizabeth can see and travel through them all. Booker closes the loop on all the universe where Comstock could have possibly arisen, but our Elizabeth is now outside those universes. She just has to pick somewhere to go.
And so, in some universe where a depressed and alcoholic Booker who has a baby daughter named Anna wakes up from a dream (as memories bleed over, remember) of another Universe, one in which he made all the wrong choices and gave up the only thing he loved and had to sacrifice everything to save her from himself, now knows what not to do (thus closing off those universes where Comstock arises which was the purpose of the sacrifice). And to remind himself that Anna is still there despite this terrible nightmare, goes to check on her. And Elizabeth gets to have the life she never had.