This a comment taken from /u/ultrafilters on reddit.com/r/math which I I think is a sufficient answer.

Generally cofinal-ness is something that appears all over the place in math, oftentimes going by the name of 'dense'. Being able to describe, or cover, a mathematical object in some minimal way can be a good first step in analyzing that object; structural properties of the smaller dense set may be easier to describe and have meaningful implications about the original object. I don't think there's anything uniquely happening in the situation you described. It just happens that density on well-founded orders (like one side of a chain) correspond exactly to density on ordinals.

The further question about "why the 'structure' of ordinals is so closely tied to their cofinality" probably doesn't have a much better answer than saying it gives us a minimal way to cover that ordinal. There are slightly better things to say about the combinatorics of ordinals as related to their cofinality using more definitions, but nothing absolutely satisfying.