The principle of non-contradiction say that p and $\neg$ p can not be both true.
- If you consider the law of excluded middle as relevant, then either p or $\neg$ p is true, and thus the other is false by the principle of non-contradiction.
- But if you consider the excluded middle as not relevant, just like intuitionistic logic, what would avoid some p and $\neg$ p to be both false ?
Actually, if true is the same than provable, then such a proposition is just a Gödel one : p is not provable and neither $\neg$ p is.
I think this is a very good interpretation and a big demystification of Gödel theorem : it is not about some true that we can not reach, it is about some absurdity that is still an absurdity when you negate it.
The statement "$p$ and $\neg p$ are both false" is formally expressed as $$\neg p \land \neg \neg p,$$ which is false by the law of non-contradiction, which states that for any $q$ the proposition $q \land \neg q$ is false (take $q = \neg p$).
More importantly, these consideration do not "demistyfy Gödel", as they refer to proposition with free propositional variables in them. Gödel constructed a sentence, i.e., a logical formula without free parameters, which is neither provable nor falsifiable (in a given formal system).
To put it another way, it is very easy to find a formula which is neither provable nor falsifiable, if we allow free variables, for example $$x < 42$$ where $x$ ranges over the natural numbers (say, in Peano arithmetic) is not provable, but neither is $$\neg (x < 42).$$