Braid

This is by far the most difficult trophy in the game.
You have to solve the entire game in under 45 minutes, this includes the final level and also the 'Epilogue' after rescuing the Princess.
Just keep going as fast as you can and don't stop running until the Timer Screen comes up

Yes, you have to collect all puzzle pieces AND put them together (when putting them together the timer will pause).

You can use the videos above to know how to solve everything, or use these speedrun videos by dphanna: