The group $S_4$ acts on the set $$X=\{\text{2-element subsets of }\{1,2,3,4\}\}.$$ That is $$X = \{\{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}\}$$
The fixed set ${\rm Fix}((1 2))=\{\{1, 2\}, \{3, 4\}\}$
$\text{Stab}(\{1, 2\})=\{e, (1 2), (3 4), (1 2)(3 4)\}$
What exactly are the fixed set and stabilizer? I've reviewed the problems in my textbook and am able to understand but I don't (at all) understand it in this context.
Why couldn't the fixed set be $\{\{1, 2\}, \{3\}, \{4\}\}$ ?
The fixed set of a group element $\sigma \in S_4$ is the subset of $X$ that is fixed by $\sigma$. The fixed set here can't include $\{ 3 \}$ or $\{ 4 \}$ because neither of those one-element sets are included in $X$. If you were considering the group's obvious action on all subsets of $\{1, 2, 3, 4 \}$, then those two subsets would be in the fixed set of $\sigma = (1~2)$.
Similarly, the stabilizer of an element of $X$ is simply the set of group elements (which will always be a subgroup) that leaves that element of $X$ fixed.