First, Freyd proves that an object in a Grothendieck category is injective if and only if it has no proper essential extensions. For each object $A$ of the category, he chooses a monomorphism $e_A\colon A\to E(A)$ which is the identity if $A$ is injective, and a proper essential extension otherwise. Then, by transfinite recursion, Freyd defines, for an object $A$, $E^0(A) = A, E^{\alpha + 1}(A) = E(E^{\alpha}(A))$ and, for a limit ordinal $\gamma, E^{\gamma}(A)$ to be a minimal essential extension of $A$ which bounds $E^{\alpha}(A)$ for all $\alpha < \gamma$ (the validity of this construct is also proved).
For a generator $G$ of the Grothendieck category $\mathsf{A}$, he considers a functor $\mathsf{Hom_A}(G,-)$ which is a faithful left exact functor into the category of $R$-modules ($R=\mathsf{End_A}(G)$). Then it is a proven that if the extension $u\colon A\to E$ is essential, so is $\mathsf{Hom_A}(G,u)\colon \mathsf{Hom_A}(G,A)\to\mathsf{Hom_A}(G,E)$.
Finally, using the fact that $R\text{-}\mathsf{Mod}$ has enough injectives, there is a monomorphism $q\colon\mathsf{Hom_A}(G,A)\to Q$ into an injective $R$-module. For any essential extension $u\colon A\to E$, $q$ then factors as $h_u\circ\mathsf{Hom_A}(G,u)$ where $h_u$ is monic as $\mathsf{Hom_A}(G,u)$ is essential (this a property of essential extensions).
The idea is the there are no more essential extensions of $A$ than there are subobjects of $Q$ which form a set was $\mathsf{A}$ is well-powered. However, I don't see why it is true. The book claims that that this is due to $\mathsf{Hom_A}(G,-)$ being faithful. The way I understand this:
The induced map from the set of essential extensions of $A$ to the set of essential extensions of $\mathsf{Hom_A}(G,A)$ is injective. However, this naive ideal doesn't take into account the fact that we work with equivalence classes (subobjects are equivalence classes). Indeed, the map on equivalence classes doesn't have to be injective as $\mathsf{Hom_A}(G,-)$ doesn't have to be full. Let $u,v$ be essential extensions of $A$ such that $\mathsf{Hom_A}(G,u)$ and $\mathsf{Hom_A}(G,v)$ are equivalent as extensions of $\mathsf{Hom_A}(G,A)$. Then there is an $R$-module isomorphism $\rho$ such that $\mathsf{Hom_A}(G,u) = \rho\circ\mathsf{Hom_A}(G,v)$. But $\rho$ is not necessarily an image of a morphism in $\mathsf{A}$.
The induced map from the set of essential extensions of $\mathsf{Hom_A}(G,A)$ into the set of subobjects of $Q$ is injective. I don't see that it is well-defined. Indeed, if $x,y$ are equivalent extensions of $\mathsf{Hom_A}(G,u)$, then there is an isomorphism $z$ such that $x = z\circ y$. Let $h_x$ and $h_y$ be respective lifts. Then $h_x\circ z\circ y = h_y\circ y$, but not necessarily $h_x\circ z = h_y$ as $y$ is not necessarily epic.
Best Answer
The idea of the proof isn't really that there are no more essential extensions of $A$ than subobjects of $Q$, but that the length of a chain of proper essential extensions of $A$ is at most the length of a chain of proper inclusions of subobjects of $Q$.
Suppose $\{E_\alpha\mid \alpha<\beta\}$ is a chain of objects of $\mathsf{A}$ indexed by an ordinal $\beta$, with $E_0=A$ and $E_\alpha<E_{\alpha'}$ a proper essential extension for $\alpha<\alpha'$. Then Freyd explicitly proves that $\mathsf{Hom_A}(G,E_\alpha)<\mathsf{Hom_A}(G,E_{\alpha'})$ is an essential extension of $R$-modules. But also it is a proper extension, since $$0\to\mathsf{Hom_A}(G,E_\alpha)\to\mathsf{Hom_A}(G,E_{\alpha'})\to\mathsf{Hom_A}(G,E_{\alpha'}/E_\alpha)$$ is exact, and $\mathsf{Hom_A}(G,E_{\alpha'}/E_\alpha)\neq0$ since $E_{\alpha'}/E_\alpha\neq0$ and $G$ is a generator.