1. Eliminating the HUD
I don't remember if the HUD of Mirror's Edge was really gone, or was it just optional. Nevertheless, my experience with the game happened entirely without it. The game is designed in such a way that it plays perfectly without one. Again, I think this works very well for immersion. Not to say that games with a HUD cannot be immersive. All attention is paid to the game world itself when there is no additional information on the screen, and this can have a huge effect on the playing experience.
One important HUD element in many games is the minimap or similar navigational device. Mirror's Edge has a different way for offering guidance: objects in the game world that lead Faith towards her goal are shown red in the otherwise white game world. Although I first considered playing without the visual aids, I pretty soon turned them back on. The gameplay experience is so much better when the player can run with some decent knowledge of where to go.
2. Tricks to increase physicality
I like to describe Mirror's Edge as the most physical video game I've played. Of course it's not physical in the same way as dancing games. It's physical in the sense that the player is given a vivid feeling of being Faith. The developers have used several neat tricks to achieve this feeling. First of all, Mirror's Edge is in true first person: the developers had done their best to put the camera into Faith's eyes. Looking down, her legs are actually visible, and of course we can see her hands actually perform all the climbing, vaulting and whatnot. They've also made the camera move in pace with Faith's step.
They've done the same with the game's soundscape. Players hear not just Faith's footsteps but also her breathing. Oh and the grunts and all the other noises. The crash of Faith hitting the pavement is one of my favorite dying sounds and made me feel really bad about screwing up. It usually goes without saying that soundscape is much more important for immersion than graphics, and the developers of ME have nailed that.
3. Simple yet powerful controls
The final piece of the puzzle are the games controls. Most of the game is played using only two analog sticks and two buttons. One button means jump, the other means crouch. These are contextualized in a natural way. Crouching while running means slide. Crouching while falling means roll. There is a high sense of mastery in playing Mirror's Edge because while things are relatively easy to do (thanks to the controls), the game never feels like it's doing things for you. Even the toughest tricks still have to be fully performed by the player.
The real genius in Mirror's Edge's design is how everything comes together to support player immersion. The designers have carefully removed everything game-y from the game in favor of a strong experience, and they have been successful. The flow of movement in Mirror's Edge feels really good, thanks to the combined effect of all the things mentioned here. I've yet to play a game that would get close to what ME does in terms of pure joy of movement. Of course, when movement is the game, it better be good.