In Goerss, Jardine book "Simplicial Homotopy Theory" at the very beginning of the book, where they prove that given a simplicial set $X$ the geometric realization is a CW-complex, there is a pushout diagram constructed $$\begin{array} A \coprod_{x \in NX_n} \partial \Delta^n & \stackrel{}{\longrightarrow} & sk_{n-1}(X) \\ \downarrow{} & & \downarrow{} \\ \coprod_{x \in NX_n}{\Delta}^n & \stackrel{}{\longrightarrow} & sk_n(X), \end{array} $$ where $NX_n$ is the set of non-degenerate simplices of $X_n$. In my head intuitively I want to believe that, $sk_n(X)$ is kind of forgetful functor that prunes $X$ from $n+1$ and upwards. For those who prefer a more rigorous definition, we can think the functor $\mathsf{sk_n}:\mathsf{Set}^{\mathsf{\Delta^{op}}} \rightarrow\mathsf{Set}^{\mathsf{\Delta^{op}}}, $ as the composition of the functors induced by the inclusion $\mathsf{\Delta}_{n \geq 0} \hookrightarrow \mathsf{\Delta}$ on the functor categories after applying $[ - , \mathsf{Set}]$, along with its induced Left Kan Extension, which turns out to be left adjoint. So this compotion gives an elegenant way to forget everything over $n$.
So, my question has to do with why/how we come up with this diagram and what's the idea behind that? Why the pushout of this diagram is the $n$-skeleta of $X$, and if this is somehow related with the functor $\mathsf{sk_{n}}$?