From nLab,
A cosimplicial object in C is similarly a functor out of the opposite category, Δ→C.
Δ here means the simplicial category.
That implies that the the identity functor on the simplicial category is a cosimplicial object in the simplicial category. That sounds really strange. Why is that?
After learning some more, I understand why simplicial objects are defined that way.
To understand it, consider a triangle, a 2-simplex. Concretely, let's regard it as a subset in $\mathbb{R}^2$. Write it as $S_{123}$.
Then consider its sub-simplices. It has three edges (1-simplex) $S_{12}, S_{13}, S_{23}$, and three vertices (0-simplex) $S_1, S_2, S_3$.
Now, collect these pieces into three sets $\Delta(2), \Delta(1), \Delta(0)$, defined by $$\Delta(2) = \{S_{123}\}$$ $$\Delta(1) = \{S_{12}, S_{13}, S_{23}\}$$ $$\Delta(0) = \{S_1, S_2, S_3\}$$
Now, we can define three "face maps" of type $\Delta(2) \to \Delta(1)$, each sending $S_{123}$ to a different face of it. Similarly, we can define two "face maps" of type $\Delta(1) \to \Delta(0)$, each sending $S_{ij}$ to a different vertex of it.
Now, we can already see how, from considering a simplex and its subsimplices, we are approaching something like the simplex category $\Delta$. So we would guess that a general simplex can be modelled by a functor $\Delta \to Set$.
However, notice that in $\Delta$, there are two arrows of type $[0] \to [1]$, but here, we constructed two arrows of type $\Delta(1) \to \Delta(0)$, so we get our clue that we should use functors out of $\Delta^{op}$ instead.
Finally, note that we have made only face maps, and still need degeneracy maps. What's a degenerate 1-simplex? That is, what's a degenerate edge? It is a vertex pretending to be an edge. There are three such edges: $S_{11}, S_{22}, S_{33}$. Add them to $\Delta(1)$.
There is only one way to construct a degenerate 1-simplex from a 0-simplex: $S_i \mapsto S_{ii}$, consequently there is only one degenerate map $\Delta(0) \to \Delta(1)$. This correspond to the one degenerate map $[1] \to [0]$ in $\Delta$.
Similarly, there are four ways to construct a degenerate 2-simplex from a 1-simplex: $S_{ij} \mapsto S_{iii}, S_{iij}, S_{ijj}, S_{jjj}$. These correspond to the four degenerate maps $[1] \to [0]$ in $\Delta$.
Now, in general, we can construct simplicial complexes using the same procedure. The result is a sequence of $\Delta(0), \Delta(1), \Delta(2)...$, each containing the 0-simplices, 1-simplices, 2-simplicies... of this simplicial complex. There would be face maps and degeneracy maps between these $\Delta(n)$, satisfying the simplicial identities.
In short, we obtain a functor $F: \Delta^{op} \to Set$. This map sends $[n]$ to $\Delta(n)$, and $f: [n] \to [m]$ to $F(f): \Delta(m) \to \Delta(n)$ defined by $F(f)(S_{v_1 v_2 ... v_m}) = S_{v_{f(1)} v_{f(2)} ... v_{f(n)}}$. Here, $S_{v_1 v_2 ... v_m}$ should be thought of as a simplex with vertices $v_1, v_2 ... v_m$.
That's why a contravariant functors out of $\Delta$ deserves the name "simplicial sets".