Let me step in where angels fear to tread. My background is in Music, Audio, and Electronics from dc to microwaves.
As I see it, one can increase the number of photons entering the camera by
a) opening the aperture wider OR
b) opening the shutter for longer.
If these are done to excess, a) has also has the effect of reducing depth of field, b) has the consequence of blurring a moving image. These two concomitant effects set the maximum number of photons that the photographer allows to enter the camera during a given shot.
The photons are passed on to the sensor and if there are not enough of them to fill it, the resultant photo will be “too dark”. Given that we have optimised the settings of time and aperture, as above, the only way to brighten the image is to apply analog gain before the sensor, or digital gain after the sensor.** As I understand it, increasing the ISO setting will add one or both of these types of gain. An analog amplifier is going to add some noise and this may show in the photo, if it is higher than the noise coming in through the lens. Whether digital multiplication adds noise, I do not know, but it presumably amplifies the quantization “noise” of the sensor.
** conversely, we have clipping if too many photons hit the sensor...
I have put my head in the lion’s mouth. Please do not chew it off, but I would welcome to have any errors of understanding or logic pointed out.
Many thanks!
David