Which of the results mentioned in the article are proven by applying some result of category theory to another field, and which are just stated using category theory (with the real mathematical insight lying elsewhere)?
There's no such thing as a theorem that can only be proven with category theory, because whatever specific thing you're applying category theory to already has all of the properties that would make the theorem true without category theory being involved - if it didn't, you wouldn't be able to treat it as if it was a category. Categories are not the kind of mathematical invention that, for example, ordinary differential equations or matrices are.
In Grothendieck's early work on abelian categories, categories are themselves the objects of mathematical study, and nontrivial results about them are established.
… makes it incomprehensible to experts in the field and laymen alike!
But sometimes it feels like Category Theory is rebuilding the world and there are genuine theorems.
Epilogue: Theorems in Category Theory