Let $F$ be a subfield of the field $K$ and $S$ be a non-empty subset (not necessarily subfield) of $K$, finite or infinite.
What kind of elements does $F\cup S$ contain?
In my notes it says that $f(s_1, \ldots , s_n)\in F\cup S$, where $f\in F[x_1, \ldots , x_n]$ and $s_1, \ldots , s_n\in S$.
Why does this hold?
The union of two sets contains exactly the things in one, or the other, or in both, sets.
The statement you've written is false: take $K=\mathbb{R}$, $F=\mathbb{Q}$, $S=\{\pi\}$, $f(x)=x+1$. Then $f\in F[x]$ and $\pi\in S$, but $f(\pi)=\pi+1\not\in F\cup S$.
Either you copied incorrectly, or there's a typo; I suspect that what is meant is that $f(s_1, . . . , s_n)$ is in the subfield generated by $F$ and $S$ - that is, the smallest subfield $G\subseteq K$ such that $F\cup S\subseteq G$. It's a good exercise to show that there really is a smallest such $G$. The statement is then true, and proving it is a good exercise.