What is the intuitive meaning of the coskeleton of a simplicial set? Consider the inclusion $i_n: \varDelta_{\leq n} \hookrightarrow \varDelta$.
This induces a truncation functor $tr_n:\mathbf{sSet}\to\mathbf{sSet}_{\leq n}$, which has a left and a right adjoint, $tr_n^L$ and $tr_n^R$, obtained via Kan extension. This allows us to define the $n$-(co)skeleton of a simplical set $X$:
$$ sk_n X := tr_n^L tr_n X, \quad cosk_n X := tr_n^Rtr_n X. $$
It is easy to see that these functors form an adjunction. 
$sk_n$ is analogous to forgetting about cells of dimension greater than $n$ in a CW complex, i.e. taking the n-skeleton. What is the dual analogy (if it exists) for the coskeleton of a simplical set $X$?
 A: For a simplicial set $X$ you can see the $n$-th skeleton to have no $m$-simplices other than the degenerate ones for $m>n$.(As you say)
For $m\leq n$ both the $n$-skeleton and the $n$-coskeleton have the same $m$-simplices as $X$.
And to answer your question for $m>n$ the $n$-coskeleton has $m$-simplices ALL possible configurations of a $m+1$ number of $(m-1)$-simplices in $\mathrm{cosk}_nX$ that agree in the appropriate $m-2$-faces to form the shape of a standard m-simplex.
If I say it in a lower dimension it will sound less tongue-twisty and easier to imagine it geometrically, which I think is what you are looking for: Take $n=2$


*

*For $m=3$ the 3-simplices of the 2-coskeleton are all four possible triangles in $X$ that you can arrange in a tetrahedral shape.

*For $m=4$ the 4-simplices are all five possible tetrahedra in $\mathrm{cosk}_2X$ that form the shape of a standard 4-simplex (pentachoron).
So in this sense, the coskeleton is the largest simplicial set containing X and the skeleton the smallest.
A: It should be added that if $X$ is a Kan complex, then $X\to \mathrm{cosk}_n X$ computes a model for the $(n-1)$-truncation of the homotopy type of $X$.  That's a fact you can easily read off using elementary definition of homotopy groups of a Kan complex, together with the bijection
$$
\{ \Delta^k \to \mathrm{cosk}_n X \} \Leftrightarrow \{ \mathrm{sk}_n \Delta^k \to X\}. 
$$
A: A simplicial set $X$ is $k$-coskeletal iff the following condition holds: 

a simplex of dimension $\geq k$ is present iff all of its $(k-1)$-dimensional faces are present in $X$. 

A standard example is the usual nerve of a small category, which is readily seen to be $2$-coskeletal. (Similarly, the nerve of an $n$-category is $(n+1)$-coskeletal).
