ISO for Analog vs Digital
ISO is confusing in digital photography in part because it was actually meant for film photography. In film photography, ISO 400 really is more sensitive than ISO 100 film.
This isn't actually true of digital photography. The sensitivity of your camera sensor is whatever it is and does not change. The sensor works a bit like an array of very tiny solar panels. Photons of light are particles that carry energy. That energy is absorbed by the photo-site on your camera's sensor array and creates a tiny charge (voltage). This is analog information -- it is not yet in digital form.
When the exposure is completed and the shutter closes, the camera will perform a read-out of the information from the sensor.
Upstream and Downstream Gain
Two things can happen and how this works will depend on the camera.
The camera can apply analog amplification. Cameras typically only do this for a handful of "stops" of gain. Since the analog information has not yet been converted to digital form, this type of gain is sometimes referred to as "upstream gain".
The analog information is converted to digital by converting voltage into digital units. This is the ADC or Analog to Digital Conversion. The output values are sometimes referred to as ADUs - short for Analog Digital Units. This is the digital output.
This digital information can also be increased by simple multiplication of the numeric values. Since this is occurring after the digital to analog conversion this is sometimes referred to as "downstream gain".
Some cameras do exclusively or mostly digital gain (downstream gain), some use a combination of both upstream and downstream gain. Since this varies by camera model there's no one right answer for how it is done.
At this point, much of the information in the digital format represents actual "signal" -- meaning this is information representing the light collected during the exposure.
But there are many interesting nuances. For example, if you power up a sensor, keep the lens covered, capture the shortest possible image, and then perform a read-out, you might think that all the pixel values would read zeros. But that's not what happens... you'll find they all read some very small values which are close to zero... but not quite zero. This represents the bias value of the sensor. BTW, modern digital cameras internally offset the bias before writing the image data.
Also if you continue to keep the lens covered but take a longer exposure, you'll find many of the pixels increase their accumulated ADUs -- even though no photons of visible light are entering the camera. There are several reasons for this and one is thermal noise (and there are camera sensors that are chilled to reduce to reduce this noise.)
There is also noise due to quantum effects. If a camera is slow to perform a read-out you can get noise caused by something called amp-glow. And while rare... the occasional high-energy photon can come wandering by and penetrate the camera.
The take-away here is that there are many causes of 'noise'.
High ISO doesn't cause noise
ABSENT from this list is ISO. ISO doesn't cause 'noise' per se. The noise captured when the exposure is complete and the camera completes the read-out is whatever it is and does not change. ISO is a gain applied after the exposure is completed (it is technically not part of the exposure - even though it's much easier to think of it as if it is part of exposure.)
Imagine a very poor quality audio recording of someone speaking ... but the microphone is too far away from the person speaking. You turn up the volume while playing the audio back and you hear lots of background noise, hiss, hum, and other nuances (maybe even including other people in the room). But because you turn up the volume, you hear all these background sounds which are much more apparent. If the microphone had been closer to the person speaking, you would have stronger "signal" and you would not have needed to increase the play-back volume.
This is a good analogy for the signal-to-noise ratio. The noise is relatively constant. It's the signal that was stronger or weaker. But this dynamic changes the "signal-to-noise ratio" (SNR).
If you have poor signal, you'll be tempted to "turn up the volume". In photography you "turn up the volume" by increasing the ISO (really the gain). But this increases ALL the information... both the signal and the noise.
Noise becomes apparent as the result of insufficient exposure
The noise simply becomes more apparent because the photograph had insufficient signal. Ultimately it is the Signal-to-Noise Ratio (SNR) that dictates how much noticeable noise you see in an image. The noise is always there ... but if the signal values are very high then the information doesn't need to be amplified so the noise isn't noticeable. Basically the signal overwhelms the noise to the point that our eyes don't notice it. If the signal is poor then we have to amplify the information. This results in a poor SNR where the noise is a decent percentage of the overall information and now we do notice the noise.
The take-away here is that the "noise" level of your camera doesn't actually change so much (it would in very long exposures where you get thermal buildup resulting in more noise). If you see "noise" in your images, it means you had insufficient signal. And since ISO isn't technically part of exposure... what it really means is you had an insufficient exposure.
What I just said is probably very contrary to what many photographers have learned. But we learn and teach photography based on the analog concepts of analog film cameras ... and we usually don't go into the depths of how digital sensors and cameras actually work. This mostly serves our needs well, but in the case of ISO and noise it creates confusion.