This is about understanding something about what goes on as you go rightward on the table at https://en.m.wikipedia.org/wiki/Homotopy_groups_of_spheres (under General Theory)

There's an applications section in the Wikipedia article, but it's all to other parts of pure math. It's hard to summarize, but they've got to do with obstructions to untangling, unwrapping, or otherwise solving things to do with spaces.

When I was working in the field it sure as heck wasn't because of practical applications. The mathematics involved is beautiful.

Mathematics has a habit throughout history of coming up with useless (to layman, at first) looking ideas that become insanely valuable. Such as zero, negative numbers, imaginary numbers, group theory and so on. Modern life and progress needs this.

Proving other theorems, which may themselves either prove further theorems or lead to direct applications. That's how the questions of "what to prove" often materialise.