While some mathematical proofs provide an explanation of why a theorem is true, this is not true of all proofs. Proof does not necessarily entail explanation.
This recent post by Conifold highlights this distinction. If one obtained a proof of the independence of Goldbach’s Conjecture (GC) from ZF (set theory), then I believe that most mainstream (Platonist) mathematicians would accept this as a proof that GC is true. Plainly, such an independence proof would provide no explanation of why GC is true. Other examples might be some proofs by exhaustion, such as the recent computer proof of the four colour theorem.
My question is, are there any philosophical schools of mathematics for which proof necessarily entails explanation?
A related question that I will sneak in here is, does refutation by counterexample necessarily provide explanation of why a theorem is false? A counterexample certainly provides a demonstration, but is that an explanation?
I’m tempted to believe that constructivism would be an example of a philosophy where proof does entail explanation, but I am not familiar enough with the constructivist philosophy to be sure. For example, would the (assumed) "non-explanatory" computer proof of the four colour theory be a constructivist proof.