Let's imagine for a second that each of those commands is a kind of little machine in a workshop or on a factory floor. It's actually a pretty good metaphor. Shell commands are generally little self-contained programs, and a program isn't too hard to think of as a machine. Each machine takes some stuff as input, chugs along for a while, and produces some stuff as output.
So you have lots of special-purpose machines at your disposal, and they do all kinds of useful stuff. In addition to the ones we already talked about, there are commands to find files, search for text and transform it, look up words, convert units and formats, scan networks, generate pictures - most of the stuff you can think of to do with a computer, really.
If you've got machines to handle a bunch of different tasks, you can get all kinds of work done. And yet... It can get awfully tedious moving things between them.
Have you ever had to save a file in one program, work on it in another, and import it into a third? Have you ever spent hours copy-and-pasting things between two programs?
I once had a temporary job that consisted of reading a name off of a sheet of paper, looking up a corresponding folder, finding a number in that folder to search for in a database program, and then using the mouse to copy and paste a different number in between two fields in the same program.
A lot of work done on computers can feel that way: Like slowly carrying buckets full of stuff in-between machines and programs.
Let's go back to our metaphorical workshop. What if all the machines had standard connectors on them for some sort of magic pipe, and the magic pipe gave you the option to move stuff between any two machines?