Let $\Omega = (\mathbb{N}_{\infty} \xrightarrow{p} \mathbb{N}_{\infty} \xrightarrow{p} \dotsc)$ be as described.
Let $S \subseteq X$ be a subobject, thus we have a bunch of compatible injections $S_i \to X_i$. Compatibility means that the diagrams
$$\begin{array}{c} X_i & \rightarrow & X_{i+1} \\ \downarrow && \downarrow \\ S_i & \rightarrow & S_{i+1} \end{array}$$
commute.
Define $\phi : X \to \Omega$ as follows: If $i \in \mathbb{N}$, we want to define $\phi_i : X_i \to \Omega_i = \mathbb{N}_{\infty}$. Well, if $x \in X_i$, then there are three cases:
$x \in S_i$ (by which I mean that $x$ lies in the image of $S_i \to X_i$). Then $\phi_i(x):=0$.
More generally, assume that the image of $x$ in $X_{i+n}$ lies in $S_{i+n}$ for some $n \geq 0$. Choose $n$ minimal. Then $\phi_i(x) := n$.
Otherwise, we define $\phi_i(x) := \infty$.
By the very construction, the diagram
$$\begin{array}{c} X_i & \rightarrow & X_{i+1} \\ \phi_i \downarrow ~~~~ && ~~~~ \downarrow \phi_{i+1} \\ \mathbb{N}_\infty & \xrightarrow{p} & \mathbb{N}_\infty \end{array}$$
commutes, i.e. $\phi : X \to \Omega$ is a morphism. One can also check that we have a pullback diagram, as desired.
Here's the idea.
The monomorphisms in the presheaf category are pointwise monomorphisms, so we can identify subobjects of a presheaf $X\in[\mathbf{A}^\text{op},\newcommand\Set{\mathbf{Set}}\Set]$ with subpresheaves of $X$, in the sense of presheaves $F$ such that $F(a)\subseteq X(a)$ for all objects $a$, and for $f:a\to a'$, $X(f)$ sends elements of $F(a')$ to $F(a)$.
Now suppose we have a subpresheaf $F$ of some presheaf $X$. We want to construct a natural transformation $X\to \newcommand\Sub{\operatorname{Sub}}\Sub(H_{-})$.
Thus for $a\in \newcommand\A{\mathbf{A}}\A$, $\alpha\in X(a)$, we need to construct a subpresheaf of $H_a$. By Yoneda, $\alpha$ corresponds to a natural transformation
$H_a\to X$, so we can just take the preimage of $F$ in $H_a$. In other words, define $G_\alpha\newcommand\into\hookrightarrow\into H_a$ by
$$G_\alpha(a') = \{ f : a'\to a \text{ such that } f^*\alpha \in F(a')\subseteq X(a')\}.$$
Then we define $\eta : X\to \Sub(H_-)$ by $\eta \alpha = G_\alpha$.
Conversely, from a natural transformation, $\eta : X\to \Sub(H_-)$, we can recover the subobject $F$ by
$$F(a) =\{ \alpha\in X(a) \text{ such that } 1_a \in (\eta_a\alpha)(a)\subseteq H_a(a)\}.$$
Side note: a subfunctor is a subobject of a functor, and a sieve is a subobject of a representable functor, but we don't really need to use these words to prove the claim.
Edit:
To see that $\eta$ is natural, let $f:a\to a'$, let $\alpha\in X(a')$.
We need to show that $\eta f^*\alpha = f^*\eta\alpha$.
Now
$$
(\eta f^*\alpha)(a'')
=
\{
g: a''\to a \text{ such that } g^*f^*\alpha \in F(a'')
\},
$$
and
$$
(f^*\eta\alpha)(a'')
=
(f_*)^{-1}((\eta\alpha)(a''))
=
\{
g: a''\to a \text{ such that } (f\circ g)^* \alpha \in F(a'')
\}.
$$
Thus, since $(f\circ g)^* = g^*f^*$, we have naturality.
Edit 2
I've been asked how we show that if we start with a natural transformation $\eta : X\to \Sub(H_-)$ and construct the associated subobject $F$ of $X$ how we show that the natural transformation $\overline{F}$ associated to $F$ is in fact $\eta$.
Let $a,a'\in \mathbf{A}$. Recall that
$$F(a) = \{ \alpha \in X(a) \text{ such that } 1_a \in \eta_a(\alpha)(a) \}.$$
We also know that if $\alpha \in X(a)$, then
$$
\overline{F}_a(\alpha)(a')
=
\{
g:a'\to a
\text{ such that }
g^*\alpha \in F(a)
\}.
$$
Putting these together we can compute
$$
\begin{aligned}
\overline{F}_a(\alpha)(a')
&=
\{
g:a'\to a
\text{ such that }
1_{a'}\in \eta_{a'}(g^*\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
1_{a'}\in g^*(\eta_{a}\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
1_{a'}\in (g_*)^{-1}(\eta_{a}\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
g\circ 1_{a'}\in (\eta_{a}\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
g\in (\eta_{a}\alpha)(a')
\}
\\
&=(\eta_a\alpha)(a').
\end{aligned}
$$
Thus as subobjects of $H_a$, we have that
$\eta_a\alpha = \overline{F}_a\alpha$, as desired.
Best Answer
If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1\to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $\mathsf{Hom}$ is continuous in its second argument, we have that $\mathsf{Hom}(-,T)\cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$\mathsf{Nat}(1,P)\cong\mathsf{Nat}(\mathsf{Hom}(-,T),P)\cong P(T)$$ where $\mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $\Omega$ are in correspondence with the elements of the set $\Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $\Omega(T)$, in this case, correspond to the closed terms of type $\Omega$ in the internal language.