What is the easiest, no-frills way to produce a simple video on Windows 8?
If you're looking for super simple and free. The Windows 8 Camera App will allow you to record video files which it then saves to the Pictures->Camera Roll folder.  It's no frills but will do the trick in a pinch.

When you open the Camera App there are two buttons one that looks like a video camera and one a still camera.  You click the video camera button and it starts recording immediately. When you're finished you click the button again and the video is saved to the location mentioned above.