$\mathcal{E}(-,E)○A$ has a left adjoint 
Theorem: If $A:\mathbf{C} \rightarrow \mathcal{E}$ is a functor from a small category $\mathbf{C}$ to a cocomplete category
$\mathcal{E}$, the functor $R$ from $\mathcal{E}$ to presheaves given
by, $$R(E):C \rightarrow Hom_{\mathcal{E}}(A(C),E)$$ has a left
adjoint $L:\hat{\mathbf{C}} \rightarrow \mathcal{E}$ defined for each
presheaf $P$ in $\hat{\mathbf{C}}$ as the colimit $$L(P)=colim(\int P \xrightarrow{\pi_{p}} \hat{\mathbf{C}} \xrightarrow{A} \mathcal{E}).$$

I'm trying to understand this theorem. What is the intuitive idea behind it?
 A: A few things: this left adjoint is, like any left adjoint, cocontinuous. Less obviously, $L$ restricts to $A$ itself, upon composing with the Yoneda embedding. This requires showing that any object is the colimit of its own representable functor's category of elements, a good thing to prove for yourself if you don't already know it. Finally, $L$ is the only possible cocontinuous functor extending $A$ in this way, which follows from the fact that every presheaf is the colimit of representables indexed by its category of elements. This is a harder but extremely important result (generalized by the theorem you've stated) sometimes called the co-Yoneda lemma.
In short, this result extends to show that cocontinuous functors out of $\widehat{\mathbf C}$ are "the same thing" as functors out of $\mathbf C$, precisely, the categories thereof are equivalent. This means the presheaf category is the free cocompletion of the category, in a way closely analogous to any free construction in classical abstract algebra--except that the presheaf category is large in size compared to the original category.
