In pure mathematics the slope stability condition is motivated only from the fact that it works. But if you are after a good intuition as for what is conceptually going on here, it helps to look at the physics analog of this, which is Douglas's "Pi-stability". This was, in turn, the inspiration for Birdgeland's general concept of stability conditions, which includes slope stability of coherent sheaves as a special case.
So, the physical interpretation of slope stability of vector bundles is revealed once one thinks of the vector bundles as being the "Chan-Paton gauge fields" on D-branes. Then the rank of the vector bundle is proportional to the mass density of a bunch of coincident D-branes, while the degree, being the Chern-class, is a measure for the RR-charge carried by the D-branes.
This reveals that the "slope of a vector bundle" is nothing but the charge density of the corresponding D-brane configuration.
Now a D-brane state is supposed to be stable if it is a "BPS-state", which is the higher dimensional generalization of the classical concept of a charged black hole being an extremal black hole in that it carries maximum charge for given mass.
Hence the stable D-branes are those which maximize their charge density, hence the "slope" of their Chan-Paton vector bundles.
The condition that every sub-bundle have smaller slope hence means that smaller branes can increase their charge density, hence their slope, by forming "bound states" into the larger, stable object.
Hence slope-stability of vector bundles/coherent sheaves is the BPS stability condition on charged D-branes.
This idea is really what underlies Michael Douglas's discussion of "Pi-stability" of D-branes, which then inspired Tom Bridgeland to his general mathematical definition, now known as Bridgeland stability, which subsumes slope-stability/mu-stability of vector bundles as a special case. But, unfortunately, this simple idea is never quite stated that explicitly in Douglas's many articles on the topic.
For more along these lines and more pointers see the discussion at
**nLab: Bridgeland stability -- As stability of BPS D-branes **
Best Answer
Here's the answer from a differential geometry perspective.
Given a unitary vector bundle $E \to M$, $M$ complex, a $\bar \partial$ operator $D: \Omega^0(E) \to \Omega^{0,1}(E)$ (where the latter is defined as sections of $\Lambda^{0,1}(M) \otimes E$) is a linear operator that satisfies the Leibniz rule on smooth forms; the space of them is called $\mathcal D$. The integrability condition says that there is a holomorphic structure on $E$ with $D$ as its $\bar \partial$ operator if and only if $D^2: \Omega^0(E) \to \Omega^{0,2}(E)$ is trivial (extending this via Leibniz to $(0,1)$-forms). (That is, the equation that $D$ come from a holomorphic structure is the flat connection equation.)
Now, the space of $\bar \partial$ operators is affine over the space $\Omega^{0,1}(\text{End}(E))$, so this is its tangent space at any operator. Also note that the group of unitary automorphisms $\mathcal U$ of $E$ acts on $\mathcal D$, such that if $D$ has $D^2 = 0$, so does $u(D)$.
What is the derivative of this action? The action sends $D \mapsto D - (Du)u^{-1}$, and noting that the Lie algebra of $\mathcal U$ is $\Omega^0(\text{End}(E))$, we see that the differential at $u \mapsto D - (Du)u^{-1}$ is $a \mapsto D a$.
Lastly let us compute the tangent space to the space of solutions $\mathcal S$ to $D^2 = 0$ in $\mathcal D$ (what we are interested in is the moduli space $\mathcal S/\mathcal G$.). This is the space of solutions of the linearization, which we compute at a solution $D$ as follows: if $d \in \Omega^{0,1}(\text{End}(E))$, then $(D+d)(D+d)\sigma = D(d\sigma) + d(D\sigma)+d^2$, and we combine $D(d\sigma) + d(D\sigma)$ into one operator, $(Dd)\sigma$. So the equation linearizes to $Dd = 0$.
Then the tangent space to the moduli space $\mathcal D/\mathcal S$, at least at an irreducible connection so that the group $\mathcal G$ acts freely, is $\text{ker}(D)/\text{im}(D)$ - the first cohomology group $H^1(\text{End}(E))$, provided we're using $D$ as our derivative operator. That is to say, it's the first cohomology of the holomorphic vector bundle $\text{End}(E)$, holomorphic structure coming from the structure induced by $D$, as desired.
Another way to understand this should come from the Narasimhan-Seshadri theorem, which I don't understand as well.