How do solid particles in the atomsphere play apart in the water cycle?
They act as condensation nuclei for water.
When water containing water vapour cools, there will a moment where the air is over-saturated with vapour, i.e. the water vapour will want to condense into liquid water. If small foreign particles, like dust is present, this condensation is a lot easier. If the air were absolutely clean, water would not condense untill well over the saturation point ("dew point").
The same goes for freezing. Clean air can cause undercooled rain, which (of course) freezes on impact.