What is the rig? It's the platform for your camera. It can be just a simple piece of aluminum or carbon fiber that you attach to your kites' line. In this simplest case, the orientation of the camera is fixed and triggering is either by a camera-internal timer or some mechanical or electronical external timer. Unless the timer triggers a regular intervals and the camera is motorized for film transport, you have to bring it down between each picture to re-arm camera and trigger mechanism.
You can progressively put more and more functionality into your rig. Most KAPers use a radio control (RC) to trigger the camera and to pan and tilt it so that it points where the photographers wants it to. Often they put the receivers' antenna on a brightly (yellow) colored stick which helps to judge the cameras orientation from the ground.
The most sophisticated constructions include a video transmitter which takes the video signal directly from the camera (if it's digital) or from a miniature camera which is mounted parallel to the photo camera. The operator on the ground has a video receiver which shows him what his camera sees. This allows a much more accurate pointing of the camera and the success rate is, at least theoretically, much higher.
I think I found the ideal display for the video feddback of my next rig: A Gameboy Advance SP. Details
Here are some pictures of my current rig: