Here's a funny thing. You are right that it's unsatisfactory because there is not going to be a satisfactory answer. The most simple explanation is it's "sticky lines in space" but that's probably even less satisfying. We can study electromagnetic waves and how they propagate, how they interact with anything, we have complex and highly accurate models for how these fundamental forces interact to make things like magnetic lines grow and stretch and interact with other things, and this skill in predicting and manipulating them is how you're reading this right now.

But it's very possible we may never know "what" they are. You cannot (as far as we know) split open a magnetic "line" and find a bunch of little guys linking arms. Or any kind of structure or new "stuff" that they can be made of.

We can work out deeper layers to reality where the waves are made of disturbances in a "field" of "something" that permeates the universe... but even that is going to hit a bedrock of our capability to understand. At a certain point when we're talking about fundamentals of the universe, there is a point we reach when asking "why" something is the way it is, where it just becomes "that's just the way it is."

A great physicist said it better than me. https://www.youtube.com/watch?v=MO0r930Sn_8