How does precipitation affect soil formation?
The process of soil formation is quite complex, so I won't be able to get into all the finer details. But here's a brief summary of why precipitation is one of the most important factors in soil deposition.
Very low precipitation (as in deserts) will prevent soil from weathering (the chemical reactions and organisms such as earthworms and bacteria depend upon water), so the result is sandy, arid soil which is very coarse and not good for plant growth.
Very high precipitation (as in rainforests) will weather the soil very thoroughly, sometimes even into clay. It will also leech nutrients from the soil, leaving it only in the upper layers; so plants that evolved to live in rainforests have shallow roots that spread over a wide area. Still, rainforests do support extremely high biomass and biodiversity despite their poor soil.
Moderate precipitation produces the best soil; it is enough to provide thorough weathering, but not enough to leech away nutrients.
Another important consideration is the flow of water once it falls onto the ground. The velocity of water flow is an important factor in the composition of soil, and depends upon the topography as well as the precipitation. Very high water velocity (as on a mountainside) stops almost all soil from being deposited at all. As water slows down (e.g. on level plains after running down a mountain), coarse sediments tend to drop out first, while finer sediments stay in suspension longer and are deposited further out.