A kinetic kill weapon is a (still theoretical) type of space-borne weapon, an alternative to exotic laser or particle beam-based systems. They work on a pretty elementary theory: why bother with a complex, heavy warhead when simple, lightweight physics will do the job for you?
In the atmosphere, you either need very accurate targeting (to make a bullet hit something), or a big warhead to hurl a shockwave and some relatively large, fast-moving debris in the direction of your target. In space, you don't really need either: all you need is a general idea of where your target is going, and then you can put a cloud of fairly small objects in its path. Objects in orbit are typically moving at immense velocities, and objects' kinetic energy is equal to 1/2 their mass times their velocity squared: Ek=1/2mv2. Even if your small objects aren't moving much at all, a lot of Ek (resulting from the target's high velocity) has to go somewhere, and it's going to go into breaking whatever your opponent worked so hard to put together and loft into orbit. But in space, there's no drag, so you don't have to worry about your small objects being slowed down or deflected by it. Why not speed them up to some ridiculous velocity and make the damage even worse? Then they can be even smaller, and you can make it even more certain that the other guy is going to run into some of them.
Here's a fun real-world analogy. Imagine standing on a highway and tossing a handful of ball bearings at an oncoming car (don't worry about getting squashed - you're the small-object deployment vehicle). The driver will see you coming, and might swerve, but if you hit him he won't be happy. Now, imagine firing the ball bearings out of a cannon. From a thousand miles away. Is he going to have time to swerve? And how will he feel about getting hit? You betcha.
Incidentally, the world's space agencies have to view all the junk orbiting our planet as kinetic kill weapons aimed at their spacecraft, which is why they're so serious about knowing where it is at all times. They also have to put armor blankets around their hardware, because even a stray paint chip can punch a hole in something important.