The Uniform Cover definition of a Uniform Space is -
A uniform space $(X,\Theta)$ is a set $X$ equipped with a distinguished family of uniform covers $\Theta$ from the set of coverings of $X$, forming a $\bf{filter}$ when ordered by $\bf {star\ refinement}$.
(I've added the bolded parts above)
What I wish to ask is $-$ why filter, and why star refinement? What is so special about these that provides sets in a cover the intuition of being of the 'same size'?
One of the models for uniform spaces is the metric spaces, and typical uniform covers are $\mathcal{U}_r:=\{B(x,r)\mid : x \in X\}$ where $r>0$. One observation we can make is that $\mathcal{U}_r \prec_\ast \mathcal{U}_{3r}$ so that such covers are a filter base under $\prec_\ast$ (star refinement). We will also want to consider other covers "uniform", but only if they're refined by some $\mathcal{U}_r$, hence we get a filter rather naturally (upward closure).
It also makes sense when comparing to the entourage formulation of uniform spaces, where one of the axiosm is that every entourage $E$ has another entourage $D$ such that $D \circ D \subseteq E$ and we also get sort of refinements when we translate this to covers in the usual way.