Scripting is simply a form of programming. That's it, there's no magic.
These days, with a modern full-featured distribution of Linux, you can manage your system pretty much only using graphical interfaces. If what you need to do is covered by those tools, then you're fine, really.
Now if some of the tasks you need to do aren't covered by these interfaces, or you need to run them in the background, while you're asleep, on a computer you only have shell access to, ... then you'll need to use something else.
And the starting point for that "something else" on Unix-type systems is the shell and shell scripting.
Your Linux system comes with a very large toolbox of utilities for your shell. Stuff for file management (copy, move, find, compare files, etc.) and text file manipulation (printing, modifying, comparing, analysis, etc.) are very common since Unix-like systems are all about files, and most software (and OS) configuration files are plain text.
Start learning those if you have a need for them. Then look into tools for whatever it is you're interested in (media conversion, database stuff, website management, system monitoring, ... the list is endless).
Backups are one of the tasks that pretty much require automation. But anything repetitive can (and probably should) be automated.
Need to convert all your FLAC files to MP3 for your media player? Script it once, you'll be able to do that in one command next time.
Need thumbnails of your latest vacation shoot to put up on your website? Script it! (And script the website update too, while you're there.)
Need to clean out old log files? Script it, you'll need to do that again next week anyway.
Need to do pretty graphs for performance metrics on your server farm? Script the data gathering and graph generation, you'll have up-to-date data whenever you need it.
(In all cases above, use existing tools to do the grunt work, your scripts should glue them all together to make them work just like you want them to.)
Knowing scripting languages (not just the shell, and not just on Linux) is just like knowing other programming languages: the more you know, the more efficient you'll be since you'll have a large toolset to pick from when you need to get something done.
There's no "hype" factor: if you've got a task to do, pick the best tool to do it out of your toolbox. On Unix-type systems, if your toolbox doesn't have a shell and a couple of scripting languages, you're missing out on a lot of the fun.
But if you can do all you need to do with graphical interfaces, then you're fine, nothing wrong with that.