You confuse true with provable, and provable and expressible.
In a particular model of a theory, statements are true or false. Having a provable statement means that we can write a proof from a certain theory. The incompleteness theorem, for example, shows that not every statement which is true in a particular model is also provable. On the other hand, if a statement is true in every model, then it is provable. The continuum hypothesis is not provable, which means that there are models where it is true, and other models where it is false. Remember that if $\sf ZFC$ have a model, then it has many different models, and not all have the same true statements.
What I mean to say is that if we assume that $\sf ZFC$ is consistent, then it is consistent that $\sf ZFC+CH$ holds, and it is consistent that $\sf ZFC+\lnot CH$ holds. It shows that $\sf ZFC$ is too weak to prove $\sf CH$. It can certainly express it.
For example, one can express it by stating what is a function, and what is an injective function. We can certainly express $\omega$ which is a countably infinite set. Then we can say "Every set $A$ which is a subset of the power set of $\omega$ either has an injection into $\omega$, or there is an injection from the power set of $\omega$ into $A$."
The point is, and I can't stress this enough, that "true" or "false" is a semantic property of a sentence. It depends on the interpretation, the model. But provability is a syntactical property which says that we can write a proof from some theory to some sentence. Syntax and semantics are related, but they are not the same.
There are theories which can prove anything which is true in a particular model, but those theories do not include $\sf ZFC$.
Let's try another example. Let's assume we have a language with $0$ and $+$, then we can write that $+$ is associative and commutative and that $0$ is the identity element.
Now consider the sentence, $\forall x\exists y(x+y=0)$. This sentence says that every element has an additive inverse. Is this sentence true in $\Bbb N$? Is it true in $\Bbb Z$? Being true is a relationship between a sentence and a model, not a sentence and a theory.
And so, if we are given a particular model of $\sf ZFC$ it has meaning to ask "Is $\sf CH$ true or false in this model?", but there is no meaning to the question "Is $\sf CH$ true in $\sf ZFC$?".
- Why is the Continuum Hypothesis (not) true?
- Impossible to prove vs neither true nor false