Category theory is basically a language. If one wanted to formulate an explicit conjecture corresponding to the original Langlands program, it could possibly be phrased as some kind of equivalence of a category of automorphic representations and a category of motives. Precisely defining those two categories is the hard part.

So even with this language, one still has to do the work of actually proving these things. It's like having a nice programming language; you still have to write the code to do the thing!