Advanced phone customization/The need to root and install a custom OS
There are many useful reasons as to why you would want to root your phone. Let's take a few:
Removing app bloatEdit
Your Andriod manufacturer would have bundled a bunch of apps on your phone, many of which you have no use for. For instance, take this image from a stock Note 3 (see left)
- you do not need the AT&T Navigator app when Google Maps is already present
- The Samsung Internet browser is a duplicate of the Google Chrome browser
And that's only a page.
The issue is that you cannot remove them from your phone. The Uninstall option does not exist. At best, you can disable them, which does remove it from the app drawer and makes it unavailable for use, but it still sits on your phone, wasting memory. And that's not true for every application: for instance, newer Samsung phones ship with a duplicate email application which cannot be disabled; it'll sit there in your app drawer always.
With a root, these 'system applications' can be easily removed permenantely from your phone.
Many Android manufacturers skin or modify their UI heavily which many people do not like. While you can reduce the problem to an extent by, for instance, using a third-party launcher, many system parts will still be skinned not to your taste. Installing a custom OS allows you to enjoy your preferred style: there are even ROMS mimicking a particular model ROM, for instance a Note 8.
In some cases, manufacturers do not configure their phone governors properly and hence your phone often does not reach its full potential. For such users, installing a custom governor (after rooting) often improves the performance. 
Most Android phones stop receiving updates after 3 years at the most, with many lower-end phones being stuck at the same Andriod version for their lives. However, custom ROM's are often available long after the manufacturer stops supporting the phone.
- Galaxy S2 stopped receiving updates by Samsung on Android 4.1. Custom ROMs are available for Android 7.
- Galaxy S6 stopped receiving updates by Samsung on Android 7. Custom ROMs are available for Android 9, on par with the newest devices.
Custom ROMs often include far more power management options that allow you to improve battery life instead of performance if you so wish. With a root, you can even downclock the cores or switch off some cores entirely (for fast phones, you may not even notice a performance reduction as modern phones can do well even with fewer cores).
- The Android app CPU throttling test mentions an example of a LG G3 wherein the stock kernel does not manage single-core loads properly, effecting a significant performance reduction. After a custom kernel was installed, the problem disappeared and the performance remained stable.