Here is a bottom view of some cycling shoes. When using cleats/clipless pedals, the plate in the bottom of the shoe clicks into a spring-loaded holder on the pedal. This happens typically by the rider simply pressing his or her foot down onto the pedal.
To remove one's feet from the pedals, typically the rider twists his or her foot to the side and the cleat pops right out of the spring loaded holder. Usually the pedal has an adjustment that can make the tension holding the foot looser or tighter.