In 'digital' terms...turning up the ISO is turning up the gain on the electric signal from the sensor.
The sensor records the light and turns that into an electronic signal. As far as I know, the lowest ISO setting on a camera is it's native signal strength...which gives you the cleanest image.
When you turn up the ISO, the camera can make an exposure with less light. In order to do that, the camera amplifies the signal from the sensor.
The problem with this is that the amplification causes distortion. This distortion is manifested as digital noise.
So the more you turn up the ISO, the more digital noise you end up with.