OK, so after a bit of researching this is what I've found.
First of all,why does the dust not settle in the water in the first place? Well, dust forms a kind of a suspension.
The particles are all similar and ionized, so therefore they have a similar charge (positive, I think) causing them repel each other and stay separate.
And because they're so small, the gravitational force acting on them isn't very strong either.
Also, the dust particles, previously dispersed in the air, have a lot of kinetic energy already.
All these 3 reasons coupled cause the particles to remain suspended for so long.
So what does adding alum do? Alum has a negative charge and tends to disperse in water very fast and very well (Why so? Because it's a colloid?).
This causes it to join up with all of the offending particles and neutralize them. Now that the particles don't have any repelling charges, they tend to clump together into 'flocs'.
The increased size as well as the lack of repelling charges cause the alum particles to settle down at the bottom or rise up and float in the water.
Why do the particles clump together and settle? Apparently even this isn't as simple as it seems.
After the particles are neutralized, they clump together because of something called the London Dispersion Force which are part of the Van der Waals forces. Basically, weak intermolecular force arising from quantum-induced instantaneous polarization multipoles in molecules causes even nonpolar particles to attract each other due to the correlated movements of the electrons in interacting molecules. More info can be found inside.
As to why they then settle, I think it can be sufficiently explained by gravity.
Oh well, that's all. All for explaining something even the ancients knew.