Amidst the set of stretching techniques labeled Proprioceptive Muscular Facilitation, there are isometric contractions. Why does isometric exercise stretch muscles?
TL; DR: It reduces the inhibitions of the muscle that produce the resistance to the stretch, allowing a further range of motion (ROM) than previously. So it doesn't really stretch the muscle, it reduces the tendency of the body to resist the stretch.
Expansion: PNF stretching is a fairly advanced, intense stretching technique that uses isometric contractions while the muscle is in a near maximal stretched state. This contraction is held for anywhere from 3-20 seconds, although one study suggests that there isn't really any difference between shorter and longer intervals in the stretch.
The mechanism is not really understood, the main theories currently is that if affects the stretch reflex, reciprocal inhibition and autogenic inhibition.
The stretch reflex is due to a muscle spindle (nerve receptor), that when a muscle is stretched suddenly, it sends a signal to the brain to contract the muscle. The greater the stretch, the more forceful the contraction. If you increase the ROM of the muscle, you are less likely to fire the stretch reflex. (As an aside, when the doctor whacks your knee with the hammer to see it twitch, that's the stretch reflex in action).
Reciprocal inhibition - Agonist/antagonist muscles are muscles in opposition to each other such as biceps/triceps, hamstrings/quadriceps, etc. When an agonist muscle contracts, the spindle sends a message to the brain for the antagonist to relax. It is suggested that this mechanism is altered by PNF stretching.
Autogenic inhibition - The golgi tendon organ fires when tension increases in the muscle, either due to contraction or to stretch, causing the agonist muscle to relax. This would override the tendency of the muscle to tighten under stress and relax, enabling a further ROM.
For additional reading, here is an writeup on PNF stretching discussing the various possible mechanisms of action, along with varying techniques of PNF (Contract/Relax, and Contract/Relax/Antagonist/Contract). It also has quite a few cited studies that are good reading. The writeup also addresses the gate-control theory which I am not familiar with.