All maps here will be (weakly) order-preserving.
Let $f:[m]\to[n]$ not be surjective. We claim there are distinct maps $g_1,g_2
:[n]\to[n]$ with $g_1\circ f\ne g_2\circ f$. Suppose $j$ is omitted from the
image of $f$. Let $g_1$ be the identity, and $g_2$ be the identity on all elements
other than $j$ and also $g_2(j)=j-1$ or $j+1$ (one of which is in $[n]$).
This works unless $n=1$, and then $m$ must be $0$, in which case we need some
ad hoc argument.
Here is some motivation for the category $sSet := Fun({\bf\Delta}^{op},Set)$ of simplicial sets, which is one of the primary uses of the category ${\bf\Delta}$.
In short, simplicial sets are meant to be a combinatorial model for spaces (up to weak homotopy equivalence), not unlike simplicial complexes. The primary difference is that simplicial sets keep track of degenerate simplices. These are pretty confusing at first, so this is what I'll focus on.
Degenerate simplices exist because of our expectations for the behavior of the morphisms in our category (of combinatorial models for spaces). For instance, surely there ought to be a morphism from the 1-simplex $\Delta^1$ to the 0-simplex $\Delta^0$! This is accommodated by simply declaring that the object $\Delta^0$ "has" a 1-simplex, which happens to be degenerate.
More broadly, if you know what all of the morphisms between simplices should be, then you know what all of the morphisms should be (since your objects are all glued together from simplices). This is where ${\bf\Delta}$ comes in: the object $[n]$ is "the universal $n$-simplex" (equipped with a total ordering of its vertices, which simplifies the theory).$^*$
This implies that the appropriate category sits fully faithfully inside of $Fun({\bf\Delta}^{op},Set)$. At the level of objects, this is because to know an object is to know its $n$-simplices for all $n$ along with the information of how they fit together: namely, the functor that it represents on ${\bf\Delta}$. At the level of morphisms, this is because a morphism should be uniquely determined by how it acts on the simplices of the source, and moreover any such function should be allowable as long as it respects the structure maps of the source -- that is, as long as it is a natural transformation.
From here, the remaining observation is that all objects of $Fun({\bf\Delta}^{op},Set)$ can be obtained in this way. Namely, a simplicial set can be viewed as a recipe for gluing together its nondegenerate simplices (or all of its simplices -- you get the same answer either way).
In case it clarifies things, let me note that above I am distinguishing between the object $[n] \in {\bf\Delta}$ and the object $\Delta^n \in sSet$. (The relationship between them is that $\Delta^n := hom_{\bf\Delta}(-,[n])$.)
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$^*$ If you think of these objects as topological simplices, then the morphisms between them are all required to be linear functions, and so are determined by their values on the vertices -- and this is what is being recorded by the morphisms in the category ${\bf\Delta}$. Said differently, there is a functor $${\bf\Delta} \xrightarrow{\Delta^\bullet_{top}} Top$$ from the simplex category to the category of topological spaces that realizes this intuition. Namely, it carries $[n]$ to the topological $n$-simplex $\Delta^n_{top}$, and it carries a morphism $[m] \xrightarrow{f} [n]$ to the unique linear function $\Delta^m_{top} \xrightarrow{\Delta^f_{top}} \Delta^n_{top}$ that carries the $i^{th}$ vertex of $\Delta^m_{top}$ to the $f(i)^{th}$ vertex of $\Delta^n_{top}$.
Best Answer
Write $[n]$ for the finite ordinal $\{ 0 \le 1 \le 2 \le \dots \le n \}$, considered as an object of the simplex category $\Delta$. At the most basic level the idea here is to think of this object as an $n$-(dimensional) simplex, and in particular to think of the points $0, 1, \dots n$ as its vertices, with the ordering giving the simplex an orientation. Order-preserving maps $[n] \to [m]$ then correspond to "degenerate subsimplices." Looking at how this works for small $n$ should help make this clearer:
You may wonder why we need these "degenerate" maps; thinking just in terms of the geometry of simplices it may make sense to restrict attention to strictly order-preserving (equivalently, injective) maps only. This can be done and gives rise to the notion of a semisimplicial set (or more generally semisimplicial object). I am not the person to ask in detail about why simplicial objects turn out to be a better thing to study.
In any case here's a rough first pass describing what a simplicial set (a presheaf on the simplex category) is: in general the category of presheaves $\widehat{C} = [C^{op}, \text{Set}]$ on a (small) category is the free cocompletion of $C$, meaning that it is obtained from $C$ by "freely adjoining colimits." So the category $\widehat{\Delta}$ of presheaves on the simplex category consists of objects obtained by "freely gluing together simplices." This gives some idea of what simplicial sets are supposed to look like as spaces (namely, they look a bit like simplicial complexes), and can be formalized using the geometric realization functor $\widehat{\Delta} \to \text{Top}$, which loosely speaking turns formal colimits of simplices into actual colimits of simplices in $\text{Top}$.
This would require a much longer answer and I'm not really the one to write it; a full answer would really require a textbook. Loosely speaking, simplicial objects turn out to be a "nonlinear" generalization of chain complexes, and can be used to write down "resolutions" of nonlinear objects such as spaces (e.g. Cech nerves) in a way that generalizes how chain complexes can be used to write down resolutions of linear objects such as modules. But this won't be a helpful answer to you until you have some experience in homological algebra (which would be a useful prerequisite for thinking about simplicial stuff).
The simplest really generally useful application of this idea is that the free abelian group on a simplicial set can be used to write down a chain complex (via the Dold-Kan correspondence); this is how singular homology is traditionally defined, using the singular simplicial set $\text{Sing}(X)$ of a topological space. $\text{Sing}$ turns out to have a left adjoint given by geometric realization, and this was one of the motivating examples of adjoint functors when Daniel Kan introduced them in 1958.