The diagram $\overline{p}$ is an $f$-colimit diagram if and only if it classifies an $f^{\prime}$-initial vertex $\Delta^0\rightarrow\mathcal{C}_{p/}$, where $f^{\prime}\colon\mathcal{C}_{p/}\rightarrow\mathcal{D}_{fp/}$ is the induced inner fibration on slices. That is to say, in the same way an ordinary colimit diagram is an initial object in the category of cocones on a given diagram, a relative colimit diagram is a relative initial object in the category of cocones on a given diagram. Thus, it should suffice to build intuition for relative initial objects.
The intuition for an initial object $x$ of $\mathcal{C}$ is that it admits an essentially unique map $x\rightarrow y$ to any other object $y$ of $\mathcal{C}$. More precisely, the mapping spaces $\mathrm{map}_{\mathcal{C}}(x,y)$ are contractible $\infty$-groupoids. The intuition for an $f$-initial object $x$ of $\mathcal{C}$ is that it admits an essentially unique map $x\rightarrow y$ over a given map $fx\rightarrow fy$ in $\mathcal{D}$ for any other object $y$ of $\mathcal{C}$. More precisely, the induced map $\mathrm{map}_{\mathcal{C}}(x,y)\rightarrow\mathrm{map}_{\mathcal{C}}(fx,fy)$ is an equivalence of $\infty$-groupoids.
Yet another equivalent characterization of an $f$-initial object $x$ is that any lifting problem
\begin{equation}
\require{AMScd}
\begin{CD}
\partial\Delta^n @>>> \mathcal{C}\\
@VVV @VVV\\
\Delta^n @>>> \mathcal{D},
\end{CD}
\end{equation}
where the upper horizontal maps takes the $\{0\}$-vertex to $x$ is solvable. In particular, if you let $\mathcal{C}_0$ denote the full subcategory of $\mathcal{C}$ spanned by the $f$-initial objects and $\mathcal{D}_0$ the full subcategory of $\mathcal{D}$ spanned by the images of the objects in $\mathcal{C}_0$, then this means the induced map $f_0\colon\mathcal{C}_0\rightarrow\mathcal{D}_0$ is a trivial fibration. This is the relative uniqueness result for $f$-initial objects. In particular, the full subcategory on $f$-initial objects lifting a given object of $\mathcal{D}$ is either empty or contractible.
I hope this illustrates why these notions are potentially useful and relatively natural generalizations of the usual colimit. Lastly, let me point out a (less trivial) generalization of the defining property: If $\overline{p}\colon K^{\triangleright}\rightarrow\mathcal{C}$ is an $f$-colimit diagram, $\overline{q}\colon K^{\triangleright}\rightarrow\mathcal{C}$ is any diagram and we set $p=\overline{p}\vert_K,\,q=\overline{q}\vert_K$, then there is a homotopy pullback diagram
\begin{equation}
\begin{CD}
\mathrm{map}_{\mathrm{Fun}(K^{\triangleright},\mathcal{C})}(\overline{p},\overline{q}) @>>> \mathrm{map}_{\mathrm{Fun}(K,\mathcal{C})}(p,q)\\
@VVV @VVV\\
\mathrm{map}_{\mathrm{Fun}(K^{\triangleright},\mathcal{D})}(f\overline{p},f\overline{q}) @>>> \mathrm{map}_{\mathrm{Fun}(K,\mathcal{D})}(fp,fq).
\end{CD}
\end{equation}
In other words, roughly speaking, natural transformations (or higher coherences between them) out of a colimit cocone are given by specifying a natural transformation of the images under $f$ and a natural transformation of the underlying diagrams in a compatible manner.
