There are two formulas involving over/under categories, that I'm trying to understand:
For a pair of functors $\mathcal F : \mathcal C \to \mathcal E, \ \mathcal K : \mathcal C \to \mathcal D$ if (pointiwse) left and right Kan extensions exist, on $d \in \mathcal D$ they are given by: $$ \mathrm{Lan}_\mathcal K \mathcal F (d) = \mathrm{colim}_{\mathcal K / d} \mathcal F p, \ \mathrm{Ran}_\mathcal K \mathcal F (d) = \mathrm{lim}_{d \backslash \mathcal K} \mathcal F p, $$ with $p$ being a projection from over/under categories to $\mathcal C$.
For a subset of morphisms $\Sigma \subset \mathcal C$ which satisfies calculus of left fractions, Gabriel and Zisman define $\mathrm{Hom}$-sets in category of fractions $\Sigma^{-1}\mathcal C$ (which is isomorphic to the localization $\mathcal C[\Sigma^{-1}]$) as: $$ \mathrm{Hom}_{\Sigma^{-1}\mathcal C}(c,c') = \mathrm{colim}_{c' \backslash \Sigma}\mathrm{Hom}_{\mathcal C}(c,p) $$
Whats the intuitive meaning of those, if we approach it from the point of view of homotopy theory ? I know only two possibly related facts: $d\backslash \mathcal K$ can be thought of as a homotopy fiber of $\mathcal K$ over $d$ and limit and colimit are $0$th cohomology and homology groups of the category with coefficients in the functor.
With Kan extensions another analogy that I have in mind is extension of group homomorphism $f:G \to H$ to the quotient $G/N$ by a normal subgroup $N$. It exists iff $N$ is in the kernel of $f$. But it looks more like a statement about cofiber rathen than fiber.
If such intuition exists, are there any textbooks or papers where it is expressed in detail ?
There is actually general machinery that produces the formula for the pointwise left Kan extension, so in some sense there is no deep meaning behind the diagram shape – it is almost forced to be what it is.
First, recall that a left Kan extension (in the original sense!) of a functor $F : \mathcal{C} \to \mathcal{E}$ along $K : \mathcal{C} \to \mathcal{D}$ is a functor $G : \mathcal{D} \to \mathcal{E}$ equipped with a bijection $$\textrm{Nat} (G, H) \cong \textrm{Nat} (F, H K)$$ natural in functors $H : \mathcal{D} \to \mathcal{E}$. In more elementary terms, this is the same as requiring a natural transformation $\eta : F \Rightarrow G K$ such that, for every functor $H : \mathcal{D} \to \mathcal{E}$ and every natural transformation $\alpha : F \Rightarrow H K$, there is a unique natural transformation $\alpha' : G \Rightarrow H$ such that $\alpha' K \bullet \eta = \alpha$.
The above definition, while simple, does not lend itself to computations and is often considered too weak to be useful. We say a left Kan extension $(G, \alpha)$ as above is:
Let us unwind the above definition of pointwise left Kan extension a bit more. Being preserved by $\textrm{Hom}_\mathcal{E} (-, E) : \mathcal{E} \to \textbf{Set}^\textrm{op}$ means $\textrm{Hom}_\mathcal{E} (G, E) : \mathcal{D}^\textrm{op} \to \textbf{Set}$ is a right Kan extension of $\textrm{Hom}_\mathcal{E} (F, E) : \mathcal{C}^\textrm{op} \to \textbf{Set}$ along $K^\textrm{op} : \mathcal{C}^\textrm{op} \to \mathcal{D}^\textrm{op}$, for every object $E$ in $\mathcal{E}$, i.e. we have bijections $$\textrm{Nat} (H, \textrm{Hom}_\mathcal{E} (G, E)) \cong \textrm{Nat} (H K, \textrm{Hom}_\mathcal{E} (F, E))$$ natural in functors $H : \mathcal{D}^\textrm{op} \to \textbf{Set}$ and $E$. In particular, this is so for representable functors $h_D : \mathcal{D}^\textrm{op} \to \textbf{Set}$, so by the Yoneda lemma, we have $$\textrm{Hom}_\mathcal{E} (G (D), E) \cong \textrm{Nat} (h_D K, \textrm{Hom}_\mathcal{E} (F, E))$$ and this is natural in $E$.
On the other hand, it is a fact that for any $X : \mathcal{C}^\textrm{op} \to \textbf{Set}$, we have an isomorphism $X \cong \varinjlim_{(C, x) : \textbf{El} (X)} h_C$, where $\textbf{El} (X)$ is the following category:
The objects are pairs $(C, x)$ where $C$ is an object in $\mathcal{C}$ and $x \in X (C)$.
The morphisms $(C', x') \to (C, x)$ are the morphisms $C' \to C$ in $\mathcal{C}$ such that the induced $X (C) \to X (C')$ sends $x$ to $x'$.
Composition and identities are inherited from $\mathcal{C}$.
In particular, $\textbf{El} (h_D K)$ has the following description:
The objects are pairs $(C, x)$ where $C$ is an object in $\mathcal{C}$ and $x : K (C) \to D$ is a morphism in $\mathcal{D}$
The morphisms $(C', x') \to (C, x)$ are morphisms $f : C' \to C$ in $\mathcal{C}$ such that $x \circ K (f) = x'$.
Composition and identities are inherited from $\mathcal{C}$.
In other words, $\textbf{El} (h_D K)$ is (isomorphic to) the comma category $(K \downarrow D)$. Hence,
$$\begin{aligned} \textrm{Hom} (G (D), E) & \cong \textrm{Nat} (h_D K, \textrm{Hom}_\mathcal{E} (F, E)) \\ & \cong \textrm{Nat} \left( {\textstyle \varinjlim_{(C, x) : (K \downarrow D)}} h_C, \textrm{Hom}_\mathcal{E} (F, E) \right) \\ & \cong {\textstyle \varprojlim_{(C, x) : (K \downarrow D)^\textrm{op}}} \textrm{Nat} (h_C, \textrm{Hom}_\mathcal{E} (F, E)) \\ & \cong {\textstyle \varprojlim_{(C, x) : (K \downarrow D)^\textrm{op}}} \textrm{Hom}_\mathcal{E} (F (C), E) \\ & \cong \textrm{Hom}_\mathcal{E} \left( {\textstyle \varinjlim_{(C, x) : (K \downarrow D)}} F (C), E \right) \\ \end{aligned}$$ and therefore $$G (D) \cong {\textstyle \varinjlim_{(C, x) : (K \downarrow D)}} F (C)$$ as you have seen.
Notice that the above formula defines $G (D)$ without saying or depending on anything about $G (D')$ for other $D'$ in $\mathcal{D}$. That is the sense in which pointwise left Kan extensions are really pointwise, and to accommodate situations where $G (D)$ does not exist for all $D$, we replace our earlier definition of pointwise left Kan extension with the above formula. That is the real story behind pointwise Kan extensions, but since it is confusing to replace definitions we often just skip the story and present the formula first.