Here is a proof of the Scholie:
For a pure Hodge structure $H$ of weight $n$ we have $H_{\mathbb{C}} = \oplus_{p+q=n} H^{p,q}$ where we have $H^{p,q} := F^p \cap \bar{F}^q$. The key point is that $H^{p,q}$ as defined above is $0$ if $p+q > n$ (but not if $p+q < n$). This is because for a pure Hodge structure of weight $n$ we have $F^p = \oplus_{p' \geq p, p' +q' = n} H^{p',q'}$ and $\bar{F}^q = \oplus_{q' \geq q, p' + q' = n} H^{p',q'}$`.
The Scholie now follows easily: It suffices to prove that $f(H^{p,q}) = 0$ for all $p+q = n$. Since $f$ is defined over $\mathbb{Q}$, $f(H^{p,q}) \subset H'^{p,q}$ and the latter is $0$ if $n > n'$.