I really don't like this, and it verges on being flat-out incorrect: the first incompleteness theorem does not say this at all. It says that there are sentences (and indeed it constructs one, the so-called Gödel Sentence) that cannot be proven or refuted within the system.
For first-order theories it follows from the contraposition of Gödel's completeness theorem that there exist (classical) models of the theory where this sentence holds and models where it doesn't: the existence of models where the sentence does not hold means that this must be a very different meaning of "true" to the one used in common parlance.
In higher-order logics, which do not have a completeness theorem, it makes sense to talk about true statements which are unprovable: there can exist tautologies, sentences which hold in all models of a theory, which cannot be proven from the axioms of a higher-order theory (and you don't need the incompleteness theorem for this); on the other hand, in a first-order theory all tautologies are provable.
Sentences which cannot be proven or refuted within a theory are said to be logically independent of the theory. Famously, the Axiom of Choice is independent of the axioms of Zermelo–Fraenkel set theory: it's up to mathematicians to decide whether they accept the Axiom of Choice. If they do, they can work in ZF+C; if not, they can work in ZF. Neither system is "more true" from a purely logical perspective, so I really don't like describing logically independent sentences as "true but unprovable": it almost certainly doesn't mean what people think it means.
The first incompleteness theorem could perhaps be stated better for a lay audience as:
No recursive set of axioms can capture our notion of arithmetic it its entirety.
Truth is subjective in sufficiently complex systems.