Spoiler warning. Please don’t read this until you’ve played Braid. It is extremely excellent and worth playing. (:
My theory is that Braid is about memory.
In World 2, it states that Tim’s memories of the Princess are fuzzy, except for the image of her braid slapping him in the face as she turned away. Tim has forgotten. He has erased his memory of what really happened.
During the game, Tim believes he is manipulating time, but in reality he is manipulating his memory. Tim claims to be seeking the Princess, but he had (or almost had) the Princess back in World 1. What he is really seeking is his memory of the Princess. The puzzle pieces are his memories. This is why the worlds become progressively darker and darker as Tim regains the puzzle pieces of his memory. He slowly begins to realize the truth about what happened in World 1 and the truth about himself that he has blocked out.
In World 5, Tim realizes that he cannot escape the shadow of his past self and his past actions.
Headless figures in World 6 symbolize that Tim is acting without a head — without his memory.
Then finally, when the last puzzle piece of Tim’s memory is obtained, Tim replays the events of World 1 in his own memory. He realizes that his memory was false and replays it in reverse — as it really happened. After fitting together all his memories, he can no longer escape the dreadful truth about himself.
There are (at least) three interpretations of the story and ending of Braid. They are interwoven, like a braid. It’s intentionally ambiguous and up to interpretation. This is my strand; I submit it as food for your thought. (: