ISO doesn't give your sensor more light. It doesn't make the sensor more sensitive. And it definitely doesn't double your exposure.
Ok, ISO is not your camera's sensor's sensitivity to light. It is a way of amplifying the signal that is recorded by the sensor while the signal is still in it's analog form.
All circuits have noise...turn the gain up on any amp or radio and it's easy to hear the noise.
Same thing for a camera. If you have a week signal, (the data captured from a camera sensor in low light conditions) then it has to be amplified to show up within the normal range, i.e. not black. That's all well and good...the only problem is, the noise, or electrical interference gets amplified also. That is the noise(digital version) or grain(film version) that you are seeing.
Modern high end cameras have ridiculously low digital noise and very sophisticated ways of telling the difference between the actual signal and the interfering noise(usually called Noise Reduction if shooting in JPEG) and can capture images at ridiculously high amplification(ISO), but if you have a very week signal and attempt to amplify it, the Signal to Noise ratio will eventually catch up and make your picture an indistinguishable bunch of speckles.
The secret to shooting with high ISO is to make sure that you aren't asking your camera to do to much...it still needs a strong signal from the sensor so that the noise won't be so apparent, i.e. enough light, a wide enough aperture, or a slow enough shutter speed.
Yes, if you amplify the signal that the sensor receives, you can, indeed, shoot with a faster shutter speed, or lower light, or a smaller aperture, thereby reducing the signal the sensor receives and still end up with an image that appears to be the correct exposure, but that isn't part of the actual exposure that the sensor sees.
In digital cameras, there are 3 parts to exposure.
1. The amount of light available.
2. How long the shutter stays open exposing the sensor to light.
3. How large the aperture opening is letting more or less light to go in.
That is the exposure triangle.
There is, however, a fourth factor in how your image will appear in print or on screen... and that is how much you amplify the signal after the sensor records its data and before it is stored on your memory card. That is ISO. Hope this clears it up.
It is relatively easy to take a high noise ISO 400 shot and a relatively low noise ISO 1600 shot. It all depends on the signal to noise ratio. That is another reason why noise usually shows up in the shadows of an image. There wasn't much of a signal there to begin with.