How does the rain mechanic work when creating a custom world?

I don't know the specifics, but I am pretty sure that it determines both the frequency and intensity of the rains you do receive. Some rains can pitter out leaving you barely damp at all, and some will leave you absolutely soaked. Unless you set this to None, you will still be getting a reasonable amount of rain in Spring.

One thing I am certain of is that rain set to Lots is constant rain.

I don't recall if frog rains are a separate setting or not.

In general, setting things to More results in them spawning reasonably frequently, and notably outside of their usual biomes, ex. gold rocks outside of rock biomes, and Lots results in them spawning pretty much everywhere to the point that you'd be hard pressed to not find them.

These observations come from my personal experience messing with the world gen settings.