As Phil V says, this subject is dependent on the camera and sensor.
I won't go into detail as it is very easy to google, just type in 'ISO invariance'. Currently AFAIK only a few sensors are ISO invariant and for some of those only in a smallish range.
This.
I own a d750 which is an ISO invariant camera, it allows a different mindset altogether.
All digital cameras only really have one ISO setting in my mind. You can change the shutter speed and/or the aperture to increase or decrease the amount of light coming in. The camera has a base ISO which is the sensors native sensitivity to that light.
Now, to effectively change the exposure beyond that native ISO you can alter it yourself in post processing, or you can let the cameras processing engine do it by the use of, usually, increasing ISO to brighten the image.
Canons, and the majority of cameras, give you a cleaner Raw file with the most information when they are exposed to the right. This extends to increasing the ISO setting in camera to get the brightest possible image in camera to maximise signal to noise ratio. (Although this hits a ceiling where they become ISO invariant too)
Conversely Nikon d750 exposed 'correctly' in camera with a given shutter speed and aperture at say iso800 is identical in image quality to the shot exposed 3 stops darker by keeping the shutter speed/aperture the same but shooting at iso100 and pushing 3 stops in your raw converter.
But what's the advantage you may ask?
Wide dynamic range scenes for a start. At ISO100 the camera is achieving its greatest dynamic range coverage. Added to that when the cameras processing engine increases exposure through you upping the ISO, it universally doubles the brightness for each stop of ISO. So your ISO800 shot of a wide dynamic range scene starts to suffer compared to your ISO100 3 stop 'under exposed' shot. Because not only does base ISO have greater dynamic range but also when you increase the brightness by 3 stops in raw conversion you can also pull back the white point and highlights selectively, giving you more usable dynamic range. The cameras processing engine does not have this finesse when upping the ISO.
Another reason and real world example is more to do with moving subjects and or handholding situations. Imagine your lens is at maximum aperture, wide open. You are shooting a moving object that will blur completely beneath a certain shutter speed. With a normal sensor you up your ISO to keep the shutter speed up.
With an ISO invariant sensor you leave it at ISO100 and record a darker image which you can push in post. Other than the dynamic range issues mentioned already you also have way more highlight headroom doing this. Upping the ISO, either in manual or aperture priority to keep the image as recorded 'bright' does in some cases risk overexposure if the light changes drastically and tricks the meter as the subject moves along. You have less of a safety net with the highlights with the higher ISO, but at ISO100 there's loads of headroom.
Downsides of shooting at base ISO with an ISO invariant sensor is dark images on the playback review.