Understanding lemma found in proof of quotient construction for DFAs
Let us start with explaining the notation:

$F$ is the set of accepting states of some DFA.
We define an equivalence relation $\approx$ on the states of the DFA as follows: $p \approx q$ if for all words $x$, $\hat\delta(p,x) \in F$ iff $\hat\delta(q,x)$, where $\hat\delta$ is the transition function of the DFA (extended from symbols to words).
We denote the equivalence class of a state $p$ by $[p]$.
$F' = \{ [p] : p \in F \}$.

Now suppose that $[p] \in F'$. According to the definition of $F'$, $[p] = [q]$ for some $q \in F$. Standard properties of equivalence relations imply that $[p] = [q]$ iff $p \approx q$. By definition of $\approx$, $\hat\delta(p,x) \in F$ iff $\hat\delta(q,x) \in F$ for all words $x$. Taking $x = \epsilon$, we see that $p = \hat\delta(p,\epsilon) \in F$ iff $q = \hat\delta(q,\epsilon) \in F$. Since $q \in F$, this shows that indeed $p \in F$.