Intuitively, I see the product-preservation or indeed finite limit preservation of geometric realization $\hat{R}: [\Delta^{op}, \mathbf{Set}] \to \mathbf{kSpace}$ as lifting (through the forgetful functor $U: \mathbf{kSpace} \to \mathbf{Set}$) a more basic left exact left adjoint $[\Delta^{op}, \mathbf{Set}] \to \mathbf{Set}$. The basic result is that such left exact left adjoints can be seen to correspond to linear orders with distinct top and bottom, aka _intervals_, and the specific interval being used for geometric realization is the standard unit interval $I = [0, 1]$. 

There is a famous result from topos theory called Diaconescu's theorem, where left exact left adjoints $[C^{op}, \mathbf{Set}] \to \mathbf{Set}$ correspond (in the sense of an equivalence of categories) to filtered colimits of representables $C \to \mathbf{Set}$. In the case $C = \Delta$, to understand filtered colimits of representables $\hom_\Delta([n], -)$, it helps to know that the category of finite intervals is dual to the category of finite nonempty linear orders (the equivalence $\Delta^{op} \to \text{FinInt}$ takes $[n] = \{0 < 1 < \ldots < n\}$ to the interval $\hom_\Delta([n], \{0 < 1\})$, with $n+2$ elements). For example, take the 
restricted representable $\hom_{\text{Int}}(i-, I)$, restricting along the subcategory $i: \text{FinInt} \hookrightarrow \text{Int}$. Like any interval, the standard unit interval $I$ can be represented as a filtered colimit of the diagram of its finite subintervals and inclusions between them, say $I = \text{colim}_{d \in D}\; I_d$ where $I_d$ is an interval with $n_d + 2$ elements. Then the functor $\hom_{\text{Int}}(i-, I)$ is a filtered colimit of representables of $\Delta$:  

$$\begin{array}{lll}
\hom_{\text{Int}}(i-, I) & = & \hom_{\text{Int}}(i-, \text{colim}_d\; I_d): \text{FinInt}^{op} \to \mathbf{Set} \\ 
 & \cong & \text{colim}_d\; \hom_{\text{Int}}(i-, I_d) \\
& \cong & \text{colim}_d \hom_\Delta([n_d], -): \Delta \to \mathbf{Set}
\end{array}$$ 

(to get to the second line, note that for any finite subinterval $J$, the functor $\hom_{\text{Int}}(J, -)$ preserves filtered colimits, i.e., $J$ is finitely presentable in the category of intervals). 

Summarizing, there are three main ingredients, all of which should be considered soft and conceptual:  

* $\Delta$ is dual to the category of finite intervals; 

* The Ind-completion of the category of finite intervals is the category of all intervals, in which finite intervals are the finitely presentable objects; 

* This implies that filtered colimits of representables $\Delta \to \mathbf{Set}$, or left exact left adjoints $[\Delta^{op}, \mathbf{Set}] \to \mathbf{Set}$, are classified by intervals, and we pick the standard interval to get (the underlying set of) geometric realization to be left exact. 

In particular, the canonical continuous map 

$$\hat{R}(\hom_\Delta(-, [m]) \times \hom_\Delta(-, [n])) \to \hat{R}(\hom_\Delta(-, [m])) \times \hat{R}(\hom_\Delta(-, [n]))$$ 

is a bijection at the underlying set level. Coupled with a brief argument that the left side is compact and the right side is Hausdorff, this map is a homeomorphism. 

Then one finishes off with a formal calculation where general simplicial sets $X$ are colimits of representables: 

$$\begin{array}{lll} 
\hat{R}(X \times Y) & \cong & \hat{R}(\text{colim}_d \; \hom(-, [n_d]) \times \text{colim}_e \; \hom(-, [n_e])) \\ 
 & \cong & \hat{R}(\text{colim}_{d, e} \; \hom(-, [n_d]) \times \hom(-, [n_e])) \\ 
 & \cong & \text{colim}_{d, e} \; \hat{R}(\hom(-, [n_d]) \times \hom(-, [n_e])) \\ 
 & \cong & \text{colim}_{d, e} \; \hat{R}(\hom(-, [n_d])) \times \hat{R}(\hom(-, [n_e])) \\ 
 & \cong & \text{colim}_d \; \hat{R}(\hom(-, [n_d])) \times \text{colim}_e \; \hat{R}(\hom(-, [n_e])) \\ 
 & \cong & \hat{R}(\text{colim}_d \; \hom(-, [n_d)) \times \hat{R}(\text{colim}_e \; \hom(-, [n_e])) \\ 
 & \cong & \hat{R}(X) \times \hat{R}(Y)
\end{array} 
$$ 

where the crucial steps are to the second and fifth lines; this is where we use the fact that the cartesian product in $\mathbf{kSpace}$ preserves colimits in each of its two arguments (which is not the case for $\mathbf{Top}$). 

Anyway, I hope the crucial role of the _order_ of combinatorial simplices is apparent here. I don't think I have a single philosophy of simplicial sets, because simplicial sets has many hearts. For example, the category of finite ordinals, including the empty one, is the initial monoidal category with a monoid; this is incredibly important for bar constructions. But for another, see this lovely <a href="http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html">blog post</a> by Tom Leinster.