The way I see it, the problems Apple aimed to address with the TB were:
(1) A vast majority of people weren't using or taking advantage of the the classical fn keys anymore (outside of system functions which are more suitable for a touch-like interface anyway), yet the fn keys occupied space on the central part of the laptop for everyone.
(2) Useful functions oftentimes have to be tucked away in submenus or the menubar because there's no great place on-screen to display them in a given situation. This is an increasing problem since both macOS and iOS get tons of new features each year, yet Apple aims to keep the systems simple and intuitive to use and features visible and accessible. But there's only so much you can add to, for example, a Finder or Pages window before you need to cram features into hidden places.
(3) Screen space is limited (especially on a laptop) and there's sometimes no good place on-screen to display useful information. For example, if you're watching a movie and want it to fill the entire screen, then obviously there can't be any controls for it on-screen because they would obstruct the movie. But that means that information like how far you are through the movie, how much time you have left will be hidden unless you wiggle around the mouse. The TB is one possible place where you can put at least some information (for example, a movie timeline can always be visible that way), though there are of course plenty other approaches to this problem aswell (using a second screen, remote apps for your iPhone or iPad, etc.).
(4) Some things or actions that you perform on your PC/Mac regularly are just more suited/intuitive to do on a Touch Screen. Moving around sliders or scrubbing back-/forward through songs or movies and the like are prime examples, but to give another example, which one do you prefer doing: hammering in the shortcut "Command" + "Z" forty times to swap through forty undo-steps, or just flicking a few times with your finger through one long undo-timeline where each undo-step is visualized? I know which one I'd prefer to do. (And yes, this undo-timeline on the Touch Bar is actually a thing on some photo editing apps, I just wish that it was more widespread and that Apple would adopt it aswell for Pages and the like)
There's probably more, but a vast majority of things that the TB does can be categorized into one of multiple of these points.
Again, this is what I think Apple views as the problems. I'm not saying that every user perceives them as problems during his everyday life, nor do I claim that the Touch Bar is a good solution to all of those, but it's certainly an attempt at addressing these issues. Just because these issues aren't always visible doesn't mean they aren't there – for example, when I'm editing a document or photo, I'm not sitting there thinking "Wow spamming a shortcut forty times to go back forty steps is so annoying", but when I saw the TB undo-feature, I thought "Well that's obviously better than the former".
