"Note that in real mathematics, we are not encountering many of those for some reason and the one Gödel provided had a self-referential relationship, and it can evidently be proved OUTSIDE of the given axiomatic system."
This is precisely the problem that non-mathematicians have trying to understand something incredibly subtle like completeness and consistency of axiomatic systems.
Gödel's theorem essentially states that in any "useful" system (although there are plenty of useful systems that are not "useful") there are things that you can say about the system that you cannot prove within that system. In particular, the statement "this system is consistent" is one of them.
You can always prove any statement, including "this system is consistent", from outside the system, by creating an axiomatic system, together with the axiom "the previous system is consistent". Well, job's a good 'un. No, of course, because your new system is also "useful", and so cannot itself be proved to be consistent without inventing another system to contain it.
Why doesn't this matter? Once you know there's a certain level of granularity you are not allowed to ask difficult questions about (remind anyone of quantum theory, as an aside?) you stop asking these questions. That's the only reason it's not of concern to the real world: because we know we won't be able to do it.
Why does it matter? Because if you want to start programming computers to do things, one of those things is to check that arguments are sound, and produce automated theorem checkers (like the inappropriately named Coq). Also, if you want to say that the human brain is just a very complicated computer, well, computers have rules they have to follow. In particular, all of our computers are Turing machines; they cannot just bullshit results. So one of two things is true: other humans and I are lying when we say we understand Gödel's theorem; or humans are not Turing machines, so cannot be emulated on a computer. They can be simulated to a high degree of accuracy, for example a third of the time just by looking asleep, but not emulated completely.