No, Gödel does not show that there are true claims S that are not provable.

He shows rather that, given a consistent formal theory T which contains enough arithmetic, then there will be a true arithmetical "Gödel sentence" G which is not provable in T. But that Gödel sentence G, though it can't be proved in T, can and will be provable in other formal theories (for example, G is provable in the theory that you get by adding to the axioms of T a new axiom Con(T) that encodes the claim that T is consistent). So if we reflect on the axioms of T and accept them as true, and so have good reason to think that T is consistent, we'll have good reason to think T's Gödel sentence G, which is provable in T + Con(T), is true. (And there's nothing especially mysterious about the notion of truth here: it is the common-or-garden notion of arithmetic truth that is invoked when we say of even the simplest sentence of formal arithmetic, as it might be "1 + 0 = 1", that it is true.)

For more -- a great deal more! -- on this, see for example my Introduction to Gödel's Theorems.