To my understanding, given a rank $r$ and degree $d$, we can fix a $C^{\infty}$ vector bundle $\mathcal{V}$ over some curve $\Sigma$ of genus $g$. We then want to study the moduli space $M_{g}(r,d)$ of holomorphic structures on this fixed bundle $\mathcal{V}$. Let $\mathcal{C}$ denote the space of holomorphic structures on $\mathcal{V}$. Naively, we would simply use $\mathcal{C} / \rm{Aut}(\mathcal{V})$ as the moduli space, but apparently this space has terrible properties, i.e. isn't even Hausdorff, etc. So I've heard Mumford's Geometric Invariant Theory is one way around this problem. And this is where you see the familiar definition of stable vector bundles: $E$ is stable if $\mu(F) < \mu(E)$ for all proper, holomorphic sub-bundles $F \subset E$, where $\mu$ is of course the slope of the bundle.
Personally, I don't have any intuition as to why this strange definition of stability leads to well-defined moduli spaces of bundles. Is there any intuition anyone can help me with, or perhaps is it just the sort of thing where you say, with hindsight, it's simply the correct thing to do to force the Geometric Invariant Theory to work?
EDIT: So restricting to simple bundles (where $\rm{Aut}\mathcal{V} = \mathbb{C}^{*}$) makes sense to me. It seems to be analogous to asking that a group action not have fixed points; thus, avoiding singularities. Maybe it would be helpful for me to consider how semi-stable bundles relate to simple bundles? (I know stable implies simple)