I would say the main relevance of oxidation states is qualitative, not quantitative. Sure enough, it is possible to give a detailed quantitative picture of redox reactions in terms of quantum chemistry without ever mentioning oxidation states, and oxidation states ultimately are an extremely simplified reflection of that picture. However, most chemists most of the time aren't at all concerned with the detailed quantum chemistry picture, because they are working at a significantly higher level of abstraction, and so the question about how an oxidation state translates to a binding energy (cf. the comments to Ivan Neretin's answer) just doesn't come up. (I see you have a Stack Overflow profile. Exaggerating the difference somewhat, I might say that such a question is akin to asking of someone writing a Java program what the program is doing with the x86 registries.)
Two accounts, already mentioned in various ways in other answers, of how oxidation states are qualitatively helpful:
Half-reactions (cf. Marcel's answer). They provide a clear picture of the key changes going on in a redox reaction, and they provide a basis for assigning oxidation and reduction potentials. Half-reactions, however, are an abstraction -- an abstraction which wouldn't be nowhere near as meaningful if they weren't interpreted in terms of oxidation states. For instance, in $\ce{3 e− + 2 H2O + MnO4- -> MnO2 + 4OH−}$ the locus of interest for someone thinking in terms of redox reactions is what is going on with the manganese atoms, and the fact that it is their oxidation state that changes, and not that of the hydrogen or oxygen atoms, reflects that. By the way, even though you don't have to use half-reactions to balance redox equations, it always was my preferred method, as it makes enough sense chemically that it makes it much harder to make silly arithmetical mistakes.
Classification of chemical species (cf. Ivan Neretin's answer). Oxidation states can give an approximate, but very useful, idea of some properties of chemical species, such as redox reactivity. For instance, -1 chlorine compounds can be nice and stable (think table salt, $\ce{Cl-}$), while +1 chlorine compounds tend to be quite strong oxidisers (think bleach, $\ce{ClO-}$). No wonder that occasionally species are named after their key oxidation state (cf. Marcel's answer).
If you want a more direct interpretation in terms of a physical property: the oxidation state loosely corresponds to electron density around an atom, relative to what it would be like if that atom was neutrally charged and isolated. You should not expect an actual formula for getting electron densities out of oxidation states (given how loose the correspondence is), nor to compare absolute electron densities around different atoms using oxidation states (given that the values are relative).
To get a better idea of what kind of correspondence I am talking about, picture a neutral chlorine atom. If an electron is added to its electron shell (thus resulting in a chloride anion, $\ce{Cl-}$, which has -1 charge), an increase in electron density happens. Now, suppose that, rather than simply getting an extra electron, the chlorine atom forms a bond with a neutral hydrogen atom, forming an $\ce{HCl}$ molecule. The situation is quite different from the first one, as the pair of electrons forming the covalent bond is shared by both atoms, in a way that can be accurately characterised in quantum chemical terms (molecular orbitals, etc.). Still, the bond is not symmetrical, but skewed towards the chlorine atom, so that the electron density around it is higher than it was in the neutral, isolated state, though lower than in the chloride anion case (and vice-versa for the hydrogen atom). The major simplification involved in assigning oxidation states is ignoring this subtlety: we say chlorine is in -1 oxidation state in both $\ce{Cl-}$ and $\ce{HCl}$. Finally, it is worth noting that adding an electron to the neutral chlorine atom promotes an electronic reconfiguration (cf. the part about permanganate in orthocresol's answer) which happens to lead to a very stable state (a noble gas electronic configuration). That explains why -1 chlorine compounds are relatively stable next to those with higher oxidation states ($\ce{Cl2}$, $\ce{ClO-}$, etc.), which tend to be powerful oxidisers.
The example illustrates how oxidation states are, from the perspective I am adopting here, an approximation (pretending all bonds are either fully symmetrical or fully skewed towards the most electronegative atom) of an approximation (using charges as proxies for electron density). Still, they are useful to the extent they:
Offer a simple model of electron transfer in redox reactions (and note that this is quite independent from all I said about electron densities: Faraday's work on electrochemistry, for instance, predates quantum mechanics and the discovery of the electron by several decades).
Can capture general trends among chemical species of an element, thus serving as basis for heuristics. Given the many layers of approximation involved, these are heuristics, and not hard rules (which explains why no one has given you hard rules here). To go beyond the heuristics, you have to look at the electronic configuration in each particular case, which, other than in very simple cases (such as $\ce{Cl-}$) is merely hinted at by the oxidation state.
