You got it right that opening up the aperture is the best method of coping with low light in most circumstances, which is why lenses with an extremely wide aperture (1.8, 1.4 or even less in rare cases) are considered low light lenses.

Changing the shutter speed can be used to compensate for low light as well of course, but it has side effects. You'll get motion blur, and if you're shooting hand-held, the image will be blurry if you cannot keep the camera steady for the duration of the shot. Of course, your camera's built in image stabilizer will reduce this, but there is a point up to which you just cannot keep the camera steady all the time anymore. Also, your motive might not hold still that long. E.g. people or leaves moving in the wind.

So basically ISO is increased to faster shutter speeds (thus less motion blur and camera shake) or close the aperture (greater depth of field) in situations in which these two cannot compensate for each other anymore.

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