I am not a physicist but let me give it a stab: except for a few specialized steps like UV or oxidizing heavy metals, most filtration is mechanical. A series of filters with smaller and smaller pores capture more and more of the mess in the water like bacteria and particulates while UV breaks down viruses, the oxidizer precipitates out metals, and so on.
None of those methods work with salt. Salts in general disassociate through ion-dipole interactions - the water dipoles essentially rip the ionic compound apart and surround each ion in what is called a hydration shell. They're bigger than bare water molecules but not much bigger - much too small to target with pore size. This shell also puts them in a thermodynamically stable state and it takes energy to "jostle" the water molecules away from the ions either through evaporation, distillation, or through another chemical reaction that precipitates out the ions.
As it turns out, doing that takes a lot of energy, so we use reverse osmosis as a cheaper alternative: we exploit the hydration shell of the ions by putting them behind a semi-permeable membrane with very small pores, "nanopores" if you will. The pores are too small for water to cross normally, but under high pressures bare water molecules can be forced through the pores while the ions trapped in their shells remain and concentrate into a brine. It takes less energy but produces a concentrated liquid waste stream that must be disposed of.
Someone please correct any mistakes I've made