Comstock begins to see Lutece as a loose end and hires his industrialist buddy Fink to have them both killed in a way that looks like an accident. Fink arranges for an equipment malfunction in the Lutece lab. Since the sabotaged machine was one of the dimensional-hopping gizmos, it doesn't actually kill the Lutece "twins" outright. Instead it maroons them in some sort of confusing state of potential death, or scattered across several realities, or whatever. The point is, the Lutece twins are now magical people that hop in and out of the story as needed and say cryptic things.
The twins figure out what Comstock is up to and decide to stop him. So, they pop over to the other universe and get Booker. Being dragged into the same universe as his other self scrambles his memories and gives the writers a handy excuse to have him know or not know anything they like. The twins send him after Elizabeth, figuring that Comstock's plans will fail without the girl.
Congratulations, it is now 1912 and we are at the very start of the game.
There are a lot of hints that this plan doesn't work on the first try. Either the twins loop through time or they just jump to other universes to grab more versions of Booker to throw at the problem. Note that when you die in the game and Elizabeth isn't around, you wake up on the floor of the Lutece office. You go through the door to find yourself right back at the battlefield where you fell. (This being BioShock Infinite's version of the Vita Chamber respawn mechanic.) It's entirely possible that every time you die, the twins grab a different Booker and dump him into the Comstock universe. Since you tend to mix memories with other versions of yourself when you change universes, this would let the fresh Booker pick up where the previous Booker left off.
Regardless of how it works, Booker is set loose in the Comstock universe. He kills dudes, rescues Elizabeth, kills more dudes, loses Elizabeth, catches her again, they make friends, they change universes a couple of times, a rebellion happens, they fight the ghost of Lady Comstock, and they look through some dimensional tears that explain the plot to them and to the audience. During all of this, there are countless references to baptism and to Wounded Knee.
(Note that the game takes place in 1912 and Wounded Knee happened in 1890. Since the age given for Booker during the game is 38, that means that Booker was 16 at Wounded Knee. That's pretty young to be in the military, but not unheard of in those days.)
Elizabeth gets recaptured again and we have a little detour where we see the future-world destroyed by fire.
She saves Booker so Booker can save her, and the two of them resume their quest. Booker finally catches up with Comstock and drowns his alter-ego in a baptismal pool. He doesn't recognize himself because Comstock is aged and has a massive beard. (And a different voice actor. I call shenanigans.)
Once the siphon is destroyed, Elizabeth can move freely between dimensions and see all the possibilities. This leads to the ending that everyone finds so baffling, where the two of you universe-hop and the game tries to fill in a bunch of the stuff I outlined above. The game is explaining the ending as it's happening, which kind of takes the punch out of it.
The idea is that there isn't just a Comstock universe and a Booker universe. Sure, we started with that, but the universes continued to split as time went on. A universe where Comstock married one woman, or another, or made peace with Fink, or became his enemy, and so on. The possibility space branched off, over and over, so that there are now "a million million" universes with Comstock. (The Booker universe branches as well, but since all he does is drink they probably all look basically the same.)
Assuming you don't want to visit a million million worlds and personally murder Comstock in each, the only way to really stop Comstock's plans is to go back to the original split: Go back to the original baptism.
This is the part of the game where people usually have trouble. Instead of going to the past and seeing young Booker in the process of baptism, you become teenage Booker, but with all of your memories intact. Then Elizabeth drowns you - or you choose it, whatever - and all of the Comstock possibilities collapse. Instead of a universe split that leads to Booker or Comstock, we now have a universe split that leads to Booker or Dead Booker.
This brings us a slight paradox: If no Comstock, then Comstock never bought Anna from Booker. This means no Elizabeth and no universe-hoping to kill Comstock. This leads to the post-credits stinger where - we assume - Booker and Anna are still together and Booker has a chance to clean up and live a normal life with her.
This explanation isn't perfect or complete, and it doesn't begin to touch on the emotional, spiritual, social, or political themes the game deals with. I know I'm still missing about 8 of the game's 80 audiologs, so there are still details to be filled in. But hopefully what we have here can help you understand the ending well enough that you can ponder the spiritual and political stuff yourself.