§ What we want:

every map $f: K \rightarrow L$ is homotopic to a simplicial map $f_\triangle: K \rightarrow L$: ie, a map that sends vertices to vertices, and sends other points through an extension of linearity. such that $f(st(v)) \subseteq st(f_\triangle(v))$ where $f_\triangle$ is the simplicial approximation of $f$.

Recall that $st(v)$ is the intersection of interiors of all simplices that contain the vertex $v$. So on a graph, it's going to be a "star shaped" region around the vertex of all the edges around the vertex.

§ Why this can't happpen

Consider a triangle as a simplex of a circle. We want to represent rotations of the circle. I can rotate around a circle once, twice, thrice, ... As many times as I want. However, if all I have is a triangle, I can represent rotating once as the map $1 \mapsto 2, 2 \mapsto 3, 3 \mapsto 1$ and rotating twice as maybe $1 \mapsto 3, 3 \mapsto 2, 2 \mapsto 1$, but that's it. I've run out of room! So I need to subdivide the simplex to get "more points" to represent this map.

§ The correct statement