It's an extremely deep topic. I have a Ph.D. in number theory (Frank Calegari was one of my classmates), so maybe that's why the article was so readable to me. Also, fortunately, it's been known for a long long time that elliptic curves with complex multiplication arise from automorphic forms, since their Galois representations are relatively easy to understand compared to general elliptic curves.

I find it a little amusing that you describe yourself as one of Frank's classmates instead, e.g, as "I'm Bill Stein".

No, they mean elliptic curves defined over number fields that don't embed into the reals.

But they don't mean the general case, do they? If I look at the abstract for the 10-author paper, they talk about CM fields.

You're right -- they do not and are instead restricting to certain specific classes of number fields. Proving some notion of modularity of all elliptic curves over all number fields is surely completely out of reach at present.