Why does atmospheric water vapor need dust particles to fall as rain?

In principle, when the concentration of water vapor exceeds the saturated vapor pressure, droplets should form. But there is a problem.

A very small droplet of water has a relatively large amount of energy stored in the surface - this is a consequence of surface tension, which has units of energy per unit area. For a small drop, the energy required to create the surface is large compared to the energy gained by water becoming a liquid - the former scales with $r^2$ while the latter scales with $r^3$, so for sufficiently small $r$ it stands to reason that the surface energy is larger than the volume energy.

This prevents a small drop from forming, and it prevents two small drops from coalescing - in order for two little drops to become a big drop, they have to go through a phase where the surface area is bigger before it gets smaller. The smaller the drops, the larger this effect. See for example this paper if you want a LOT of detail, or also this description - a lot more accessible. From all this we conclude it's hard for a big drop to form spontaneously. So you need "something" that starts the process - a nucleus that helps overcome the surface tension term, and that allows a drop to get past that critical small stage.

And that's why it is much easier for vapor to condense into small droplets in the presence of small particles.

You might be interested in this answer I wrote earlier on a related topic - namely the formation of haze in high humidity areas with and without particulate contamination.