Limbo is another good platformer, it's fairly short like braid and you'll die a lot but it's very forgiving, mainly placing you to right before the moment you died.
A few more suggestions are:
Fez:
Similar to braid, it's a very well done puzzle platformer with a unique gameplay mechanic.
In braid it was time travel, in fez it's warping the perspective of the level to reach places you couldn't.
It has a heavy pixel art style and lasts longer than braid just because of the crazy number of secrets hidden everywhere.
The level warping affect might feel strange but after 20-30 minutes your brain will just accept it and you'll be able to move along.
Antichamber:
I'd consider this a mix of braid meets open-world portal.
It's first person, but each area has its own little mechanics you'll need to learn to get through it.
Don't worry about the harsh looking visual style, it was much less than I thought once I started playing.
Portal:
Since I've mentioned it, portal & portal 2 for obvious reasons. Often considered "THE" first person puzzle game.
Hyper light drifter:
An unusual one, I'm putting it here not because of the gameplay, but because of the art and story.
This is an atmospheric top down exploration game with short difficult fast paced combat segments that are very forgiving, doesn't matter if you die you can just try again or walk away.
The art style is super vivid pixel art and its story is just like braid where you don't quite get it until the end then you'll end up watching an explanation video on youtube and preceed to go "Aah" for the next week.