Exactly. An "explain like I'm twelve" for Godel's Incompleteness Theorum itself would revolve around the idea of self-referential statements (such as 'the set of all sets which don't contain themselves', or 'the barber of Seville shaves everybody who doesn't shave themselves').

In my understanding, Godel created a system that mapped statements to numbers, and then looked at the numbers that represented statements like 'this statement is provable' and found a way to show that the equivalent number had a property that wasn't provable.

Is that so? I think you mean that this is an explanation of the theorem, but not of a proof of the theorem.