Most star systems are binary, but why is that? Why would new stars form close to others, and not (relatively) evenly spread? And even if they are clustered close together, why are the majority binary? Or are binaries the ones that survive, and systems with 3 or more stars tend to be unstable and quickly collide?
If that is so, what makes binary systems stable? Why do the stars not simply spiral in and collide, or break away shortly after creation?
Collapsing gas clouds fragment into multiple cores because the Jeans mass, that determines the minimum mass that becomes gravitationally unstable to collapse, becomes smaller if the cloud is able to contract without heating up too much. i.e. $$M_J \propto T^{3/2} \rho^{-1/2},$$ where $\rho$ is the cloud density. Thus if the cloud density can increase but the temperature stays constant(ish), then the Jeans mass shrinks and the cloud becomes unstable to further fragmentation.
Multiple systems with $n>2$ are inherently unstable unless they are hierarchical. i.e. a star in a wide orbit around a close pair can be stable, as could two close binary systems orbiting each other. The condition for stability is roughly that the separation of the wider star must be 5-10 times that of the inner pair ([dependent on mass ratios and eccentricities - Eggleton & Kiseleva 1995). In most other cases, what happens is the ejection of other stars from a multiple system, leaving behind a binary system Durisen et al. (2001).
What makes a binary stable? We'll what would make it unstable? I'm not sure why you think they should spiral in towards each other. This can't happen unless there is some dissipative mechanism, like tidal interactions. Gravitational waves are ineffective in all but close binaries involving compact stellar remnants. They can't "break away" because they are gravitationally bound.
EDIT: Note that it isn't true that most systems are binaries. The majority of "systems" are in fact single stars. The binary frequency for solar-type stars is about 50% - i.e. as many singles as binaries; but the binary frequency for the much more numerous M-dwarfs is probably around 30% and so single stars outnumber binary systems, though it is not clear whether that is true at birth Duchene & Kraus (2013).
Two massive bodies orbiting each other form stable orbits. This is called the "two-body problem." Add a 3rd body to the system and the results are unstable orbits. It's akin to the motion of a singe pendulum. A single pendulum swinging back and forth is a linear system with motion that is repeating and predictable. Similarly two bodies orbiting each other is a linear system. A double pendulum is a non-linear system where the motion is sensitive to initial conditions. The resulting motion is "chaotic." Similarly, a three body system for orbits is inherently chaotic. In nature, third bodies that form close in to two other bodies are likely to get ejected from the system so you are just left with two bodies. You can see all this for yourself if you play Super Planet Crash http://www.stefanom.org/spc/
After what I recall from astronomy class ... for the gas cloud to form a star it has to overcome some "difficulties" - forces that restrain further shrinking. In the first place the molecules need to move very slowly .. only cold clouds can become stars - otherwise they couldn't form regions with higher mass concentration that would start the entire shrinking process. Next problem is angular momentum. Due to shrinking and molecules arriving out of center to the shrinking core (which later will become the star) - they induce a rotation - at some point the rotation would limit further shrinking of the gas cloud by centrifugal force. Hence the cloud needs a mechanism to loose angular momentum .. which is done by friction against the remaining material - but this takes a long time. Models show that by building 2 cores that circle each other this momentum is roughly halved and shared between those cores - Which allow faster shrinking of the gas cloud. Add in what @Kyle says about stability of 2 body systems vs 3 body systems and you get your predominance of binary star systems
