Confusion about the application of Hahn-Banach extension Theorem in a duality argument

Question

in my notes I have the following discussion.

Let $V \subset H$ two Hilbert spaces, with $V$ continuously embedded in $H$ and let us identify $H$ with its dual space. Then, for $f \in H$, we write $\tilde{f}$ for the Hahn-Banach extension of $f$ to $V'$ $$<\tilde{f}, v> = (f,v)_{H} \quad \forall v \in V \quad (\star)$$

I cannot understand two things:

1. How is it possible to extend an element of $H$ to $V'$? I mean, since $f \in H=H'$ I can apply Hahn-Banach, but I really cannot see why I can extend to $V'$, since $V \subset H$.

2. How is that definition $(\star)$ linked to the fact that $\tilde{f}$ is the extension of $f$ to $V'$? From that definition I can only say that the map $\tilde{f}: V \rightarrow \mathbb{R}$ such that $v \mapsto <\tilde{f},v> = (f,v)_H$ is continuous and hence $\tilde{f} \in V'$