Why does Raoult's law work?
Raoult's law is an approximation for ideal solutions.

Those links have more detail, but the basic idea is that when you assume that the interactions between solute and solvent molecules don't depend on the identity of the molecules, but the vapor pressure does, then it makes sense that the vapor pressure of the mixture would be a linear combination of the vapor pressure of the molecules in the mixture.

Think about it like a bunch of slightly sticky spheres bouncing around in a bucket. A certain percentage will randomly bounce out of the bucket - that's the vapor pressure. If you have a different type of sphere with less mass, more will bounce out in the same amount of time - the vapor pressure will be higher. If you mix them, the total vapor pressure is given by the total number of both types that bounce out. The more volatile ones leave more easily, so the total increases.

Since ideal gases don't interact with each other and since ideal mixtures have negligible enthalpy of solvation, all that is left is entropy terms. If you are looking for a connection to entropy, that is it.

In cases where the ideal approximation is not realistic (water and a volatile solute, for example), Raoult's law doesn't really hold, but it is still a reasonable approximation when the concentration is low simply because most functions look approximately linear for small deviations. It also tends to work reasonably well because most things that will form a solution tend to have similar interaction energies - otherwise they are less likely to mix in the first place.

Of course, there are many, many practical cases where Raoult's law doesn't work, and so we have corrections for that: fugacity and activity.