On my first intro flight. I noticed the instructor twisting the mixture knob. I didn't think of asking, but now Im curious.
It's called a Vernier Control. It has an internal screw that is part of the fixed housing, with threaded jaws that engage the threads around the outside and are part of the knob.
When you turn the knob, you are turning the jaws with internal threads like a nut, and the whole thing moves in when you turn clockwise (richer) and out when counterclockwise (leaner), moving the teleflex cable running to the carb or fuel injection controller with it. When you push the center button, it opens the jaws, disengaging the threads and now the knob can be pushed straight in and out like the throttle.
It makes it easy to make very fine adjustments by turning the knob, but when you need to make large changes, you don't have to furiously spin the knob, you just push the center button in with your thumb and you can pull the whole thing in or out. In an emergency when you need to go "balls to the wall", just shoving the knob forward with the heel of your hand pushes the button in automatically and lets you move it in one motion.
You usually see mixture and propeller controls done this way, because most of the time you are making fine adjustments until those times you need to shove them all the way in or all the way out. You'll sometimes see Vernier throttles as well, but it's less common.