1- Why did he require $V \cap E$ to be nonempty? (If $x$ is in $E$ or $x$ is a limit point of $E$, then $V \cap E$ is always nonempty.
2- Why didn't he require $x$ to be a limit point of $E$? What if $x$ were isolated?
(In the general definition of a limit on metric spaces, $x$ has to be a limit point of $E$)
It has to do with the empty set: any predicate of $z$ that follows "$\forall z \in \varnothing$" makes the proposition true. Suppose you want to take the limit of $f$ as $t\to x$, but $x$ is not an accumulation point of $E$. Then there exists at least a neighborhood of $x$ that does not intersect $E$, which means that the condition $$\exists A\ \mathrm{s.t.}\quad \forall U \in \mathcal U(A) \quad \exists V \in \mathcal U(x)\ \mathrm{s.t.} \quad \forall t \in V \cap E \setminus \{x\} \qquad f(t) \in U $$ would always be true because there are no $t$'s in $V \cap E \setminus \{x\}$. That would imply that any $A$ would work and would render the definition of limit quite useless.
So that is why he requires $V \cap E \neq \varnothing$.