This is from the 2nd statement of proposition $4$ from Dummit & Foote.
It says the second is proved similarly, how so? If the argument is the same, doesn't it show $F_1 \subset F_2$
Because $H_1 = \{ \sigma(f_1) = f_1 \} \leq H_2 \leq Aut(K)$. So anything that is fixed in $H_1$ is fixed by $H_2$. Why do we also use field extension to denote subgroups?
Given $x \in F_2$ then by definition $x$ is fixed by every element of $H_2$. Since $H_1 \subseteq H_2$, then $x$ is also fixed by every element of $H_1$, so $x \in F_1$.
Since $H_2$ is the larger subgroup, being fixed by every automorphism in $H_2$ is a stronger condition than being fixed by every automorphism in $H_1$. Correspondingly, the set of elements fixed by $H_2$ will be smaller than the set of elements fixed by $H_1$.