This is not much of an answer, but it might help. The category of simplicial sets should be thought of as the free cocompletion of $\Delta$. In other words, it's what you get if you freely take colimits of simplices, and since taking colimits is the most general form of gluing, it's precisely the most general setting for gluing simplices abstractly.
The essentially geometric nature of this construction is maybe clearer if you first restrict to the subcategory of $\Delta$ on the first few objects. For example, on the first two objects you get the category of graphs.
Anyway, this immediately implies the existence of geometric realization $\text{sSet} \to \text{Top}$, which is given by interpreting an abstract colimit of simplices as a topological colimit of (geometric realizations of) simplices. If you know that every topological space is weak homotopy equivalent to a CW-complex, maybe the idea of relating the two categories by geometric realization is not so strange.
Edit: You may also be under the impression that $\Delta$ is somehow unique with regard to this property, which is not true; see the nLab article geometric shape for higher structures. Simplicial methods and homotopy are closely related to higher categories via the homotopy hypothesis and there are several related ways to approach this; the model category structure is just a shadow of the higher-categorical structure.
Edit #2: I should mention that there is a sense in which $\Delta$ is special. To get a test category of simplices you'd like to be able to take products of low-dimensional things to get higher-dimensional things, but you'd also like to be able to project from high dimensions to low dimensions The laziest way to do this is the universal one, namely: take the free monoidal category on a monoid. (Edit:) This gives you the augmented simplicial category, which is the category of all finite ordinals and ordinal-preserving maps, and for reasons I don't completely understand we remove the empty ordinal to get $\Delta$. Whether this is directly responsible for the success of $\Delta$ in homotopy theory I can't say.