[I wrote this yesterday, but didn't post it]
Let's assume that mathematics is a job that follows the following rules: You use two languages: Predicate Logic and Natural Deduction. You use Predicate Logic to write claims like "There are infinitely many prime numbers", and you use Natural Deduction to construct proofs of such claims. Both languages are formal. And you can even use a computer to verify that you're using them correctly.
To prove things, you need to assume that some things are true. These "assumptions" are called axioms, and are expressed using Predicate Logic. Philosophically speaking, the rules of Natural Deduction can be thought of as axioms for how to correctly construct proofs, but they're usually called "inference rules" and not axioms for minor technical reasons.
The methodology which we described arises inevitably from a proof-driven approach to mathematics, because mathematicians need a way to agree which methods of argumentation they think are correct, and also to prevent people from making things up. Admittedly, this is only true for those mathematicians who have looked into these foundational matters.
Some axioms can lead to an accident where you can prove a proposition and its negation simultaneously. (An example in set-theoretic foundations is Unrestricted Comprehension.) By a simple result called the principle of explosion, in such a situation you can prove every proposition and its negation. So such axiom sets are useless. We call them inconsistent.
A question arises: Given a set of axioms which is sufficient to serve as a foundation of mathematics - like ZFC set theory for instance, or less obviously Peano arithmetic - is it possible to construct a proof of their own consistency from those same axioms alone? The answer is no, and this is Goedel's 2nd Incompleteness Theorem.
You can - in a way - prove that Peano arithmetic (for instance) is consistent, but you need to assume axioms which are strictly more powerful than Peano arithmetic. And to show the consistency of those axioms, you need to assume yet stronger ones. This is quite a disturbing situation.
Ultimately, the theorem is a statement about the methodology for doing mathematics which we described. But there is no escaping it if mathematics is about stating claims and proving them.