Let me concentrate on your first question; this should clarify the authors' claim. We'll see if that's enough for you to figure out the rest.
Even though later we will be interested in the rather specific $k$-vector space $\hom(V,W)$ of linear maps, for now, it is conceptually easier to consider any finite-dimensional $k$-vector space $V$. I like to think of it as a vector bundle over $\mathrm{Spec}(k)$. And a vector bundle (thought of as a sheaf) is ought to have a "total space" – a scheme $|V|$ over $k$ whose sections correspond to the elements of $V$, universally. Meaning that for every $k$-scheme $X$, the $k$-morphisms $X\to |V|$, being the same as the sections of the pull-back $|V|\times_kX$, should be the global sections of the pulled back vector bundle $V\otimes_k\mathcal{O}_X$, i.e., $V\otimes_k\mathcal{O}_X(X)$. For short, we want $\hom_k(X,|V|) = V\otimes_k\mathcal{O}_X(X)$.
I claim that this is solved by $\mathrm{Spec}(S^\bullet V^\vee)$. In fact,
$$\begin{align*}
\hom_k(X, \mathrm{Spec}(S^\bullet V^\vee))&=\hom_{k\text{-alg}}(S^\bullet V^\vee,\mathcal{O}_X(X))\\
&\cong\hom_{k\text{-vect}}(V^\vee,\mathcal O_X(X))\\
&\cong V\otimes_k\mathcal O_X(X),
\end{align*}$$
where the bottom isomorphism comes from the natural map $V\otimes_k\mathcal O_X(X)\to \hom_{k\text{-vect}}(V^\vee,\mathcal O_X(X))$, mapping a homogeneous element $v\otimes f$ to the homomorphism $(\varphi\mapsto \varphi(v)\cdot f)\in \hom_{k\text{-vect}}(V^\vee,\mathcal O_X(X))$. It's an isomorphism since $V$ is finite-dimensional.
Returning to $\hom(V,W)$ and its associated affine scheme $|\hom(V,W)| = \mathrm{Spec}(S^\bullet \hom(V,W)^\vee)$: Let $U\subset\hom(V,W)$ be the affine subspace consisting of those linear maps $V\to W$ which restrict to the identity on $W$; equivalently, "that split the inclusion map $W\subset V$". Moreover, for every $k$-algebra $\mathcal O_X(X)$ it makes sense to define $U\otimes_k\mathcal O_X(X)\subset \hom(V,W)\otimes_k \mathcal O_X(X)$ in an obvious way and there exists an affine sub-scheme $\mathcal U\subset |\hom(V,W)|$ such that via the above isomorphisms, $\hom(X,\mathcal U) = U\otimes_k \mathcal O_X(X)$. (I'll leave the details up to you.)
What the authors claim is simply that via the indicated map $\mathcal G_W\to |\hom(V,W)|$, $\mathcal G_W$ is isomorphic to $\mathcal U$. (Let me know in the comments if you need more clarifications or more hints towards the proof.)
$\newcommand{\D}{\mathfrak{D}}\newcommand{\set}{\mathsf{Set}}\newcommand{\psh}{\operatorname{Psh}}\newcommand{\H}{\mathsf{H}}$I will use my own notation out of preference, but everything I say here is easily translated to your notation.
Let $\D$ be a locally small category, and $\H:\D\to\psh_\D$ the Yoneda embedding. I claim that an arrow $U\overset{\alpha}{\longrightarrow}V$ in $\D$ has $\H(\alpha):\H(U)\implies\H(V)$ an epimorphism in $\psh_\D$ if and only if $\alpha$ is a split epimorphism.
Exercise $6.2.20$ in Leinster's basic category theory ask one to show that, if the codomain has all pullbacks, arrows in functor categories (=natural transformations) are mono/epi iff. every single one of their components is. I will give or sketch my solution to this exercise if you wish. Suffice it to say, $\H(\alpha)$ is an epimorphism iff. every component is.
Note: Leinster phrases the result for small domains and locally small codomains only. In private communication with him, he said that most of his insistence on small categories was to avoid foundational quibbles in his book, which was to be as simple as possible. However, if you don't mind the possibility that $\psh_\D$ is not locally small, then don't worry!
That is to say, if and only if every $\D(A,U)\overset{\alpha\,\circ\,-}{\longrightarrow}\D(A,V)$ is an epimorphism in $\set$, i.e. a surjection, for all $A\in\D$. That is true if and only if every arrow $A\to V$ factors via $A\longrightarrow U\overset{\alpha}{\longrightarrow}V$.
Suppose $\alpha$ is a split epimorphism: by definition, that is iff. there is a 'section' $\beta:V\to U$ that $\alpha\circ\beta=1_V$. Then given $f:A\to V$, I can define $g=\beta\circ f:A\to U$: $\alpha\circ g=f$ is a desired factorisation.
Conversely, suppose $\alpha$ has this property for all $A$. I can then specify $A=V$, and say that the identity $1_V$ factors through $\alpha$. That means there is $\beta:V\to U$ that $\alpha\circ\beta=1_V$, i.e. $\alpha$ is a split epimorphism.
In conclusion, $\H(\alpha)$ is an epimorphism of $\psh_\D$ iff. $\alpha$ is a split epimorphism. A little whimsically, if your category satisfies the axiom of choice, then $\H$ indeed preserves epimorphisms.
Best Answer
Let $\mathcal{C}$ be a small site (or, at least, cofinally small) and let $\textbf{Psh} (\mathcal{C})$ be the category of presheaves on $\mathcal{C}$. There is a functor $\Gamma : \textbf{Psh} (\mathcal{C}) \to \textbf{Set}$ represented by the terminal presheaf (which may or may not be representable in $\mathcal{C}$, at this level of generality), and it has a left adjoint $\Delta : \textbf{Set} \to \textbf{Psh} (\mathcal{C})$ that sends every set $A$ to the "constant" presheaf defined by $(\Delta A) (U) = A$. We have a counit morphism $\epsilon_F : \Delta \Gamma F \to F$ for every presheaf $F$, and your construction is precisely the image of this morphism. Expressed this way, the failure of $\operatorname{Im} \epsilon_F \subseteq F$ to be a sheaf becomes unsurprising: usually we have to sheafify the presheaf image to obtain a sheaf.
If your goal is to construct a flabby (pre)sheaf, then it would be inappropriate to sheafify $\operatorname{Im} \epsilon_F$: sheafification can destroy flabbiness. On the other hand, if we work with presheaves then representability is a rather strong condition: indeed, representable presheaves are projective, so the epimorphism $\Delta \Gamma F \to \operatorname{Im} \epsilon_F$ would be split. But that would make $\operatorname{Im} \epsilon_F$ a retract of a constant presheaf, hence also constant – not very interesting, I think.
Finally, let me remark that the notion of flabby (pre)sheaf does not seem to be appropriate for non-localic sites. The point of flabby sheaves of modules on a topological space or locale is that they are acyclic with respect to the global sections functor, but I don't think this is true for a general site.