Think of it this way: Let $V$ be a model of $ZFC_2$. Then I claim CH holds in $V$ if and only if $CH$ is actually true (note that in order for second-order logic to make sense, we have to make a commitment to an underlying "real" universe of sets). The proof of this is as follows. First, $\omega^V$ has order type $\omega$: clearly it has a subset of order type $\omega$, and by the second-order version of the powerset axiom, $P^V(\omega^V)=P(\omega^V)$, so if $\omega^V$ had the wrong order type $V$ would "see" the error. A fortiriori, we can deduce that $\omega^V$ is countable.
By similar reasoning, $P^V(P^V(\omega^V))=P(P(\omega^V))$. Now CH is false if and only if $P(P(\omega^V))$ contains three infinite sets $X, Y, Z$ no two of which have the same cardinalities (left-to-right is trivial; right-to-left follows from the countability of $\omega^V$).
Suppose $CH$ is false; let $X, Y, Z$ be as above. Since $P(P(\omega^V)=P^V(P^V(\omega^V))$, we have $X, Y, Z\in V$; by the axiom of extensionality, $V$ sees that the cardinalities of $X$, $Y$, and $Z$ are different, and by the second-order powerset axiom $V$ sees that $X$, $Y$, and $Z$ are infinite. So $CH\implies (ZFC_2\models \neg CH)$.
Suppose now that $CH$ is true. Let $X, Y, Z\in P(P(\omega^V))$; again, we have $X, Y, Z\in V$. Since $CH$ holds, by the second-order powerset axiom plus separation we can find a bijection $f$ between two of $X, Y, Z$, so $CH$ holds in $V$. So $\neg CH\implies (ZFC_2\models CH)$.
This shows that $ZFC_2\models CH$ or $ZFC_2\models \neg CH$. The point is that the full power of second-order logic allows $V$ to "ask" certain set-theoretic questions of the "real" underlying universe of sets; these questions include ``Is CH true?" Similarly, it seems to me that they include all questions of the form "Does $V_\alpha\models \phi$ hold?" where $\alpha$ is a computable ordinal and $\phi$ is $\Sigma_1$ over $V_\alpha$ ($\Sigma_1$ is somewhat arbitrary; higher quantifier depth can (I believe) be achieved by passing to larger computable $\alpha$).
I'd imagine that in fact this phenomenon extends much further than what I've outlined, and that a staggeringly large class of sentences of set theory are known to be decided in $ZFC_2$, even if we don't know which way they are decided.
