For $n \in \mathbb{N}$, the standard simplicial $n$-simplex $\Delta[n]$ is the simplicial set which is represented (as a presheaf) by the object $[n]$ in the simplex category, so $\Delta[n]= \Delta(-,[n])$.
Very often it is said that standard $n$-simplex can be pictured like this (for small $n$):
But I don't understand it. It doesn't look like a realization functor from simplicial sets to topological spaces. And standard $n$-simplex as a simplicial set has $\omega$ many non-empty sets of $k$-simplices, not finitely many.
The same goes for pictures of $k^{th}$ horns:
So my question is why $n$-simplex or horn can be depicted in such way?