I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \subseteq \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that:
$$ A = \{x : (x,a) \in U \} $$
Having constructed $U$, we can construct a new universal set $V$ that is $\Sigma_{\alpha+1}^0$. To construct $V$, Jech uses some continuous mapping of $\mathcal{N}$ onto the product space $\mathcal{N}^{\omega}$ such that:
$$ (x,y) \in V \text{ if and only if for some $n$, } (x,y_{(n)}) \not \in U $$
where $y_{(n)}$ represents the $n$th coordinate of the image of $y$ in the product space $\mathcal{N}^{\omega}$ under the continuous mapping, which can be some continuous pairing function $\Gamma$, i.e. for instance, we can have $y_{(n)}(k)=y(\Gamma(n,k))$.
I would like to get some help on the intuition behind having to use a continuous mapping to construct $V$. From what I understand, $(x,y) \in V$ if and only if $x \in \bigcup A_n$ if and only if $(x,y) \not \in U$ (where the $A_n$'s here are $\Pi_{\alpha}^0$ sets) ... Why not just take $(\mathcal{N} \times \mathcal{N}) - U$ to form $V$ ?
For $a\in \mathcal{N}$ and $X\subseteq \mathcal{N}\times \mathcal{N}$, let's define $X_a = \{x\mid (x,a)\in X\}$. In particular, $U_a$ is the $\Sigma^0_\alpha$ set coded by $a$.
Now let's follow your proposal and define $V' = (\mathcal{N}\times\mathcal{N})\setminus U$. Note that $V'$ is the complement of a $\Sigma^0_\alpha$ set, so it is $\Pi^0_\alpha$.
For all $a\in \mathcal{N}$, we have $b\in V'_a$ iff $(b,a)\in V$ iff $(b,a)\notin U$ iff $b\notin U_a$. So $V'_a = \mathcal{N}\setminus U_a$. Since every $\Pi^0_\alpha$ set is the complement of some $\Sigma^0_\alpha$ set of the form $U_a$, we see that $V'$ is a universal $\Pi^0_\alpha$ set.
But the goal here is to find a universal $\Sigma^0_{\alpha+1}$ set, not a universal $\Pi^0_\alpha$ set! So we want to find a $\Sigma^0_{\alpha+1}$ set $V$ such that the sets of the form $V_a$ are arbitrary countable unions of sets of the form $V'_x$. This is the motivation for mapping $\mathcal{N}\to \mathcal{N}^\omega$, so that a single $a\in \mathcal{N}$ can code a countable sequence of sets of the form $V'_{a_{(n)}}$.