In my understanding, the phrase "the rain let up" means the rain became less severe or stopped, but it doesn't sound very much idiomatic, so what are some alternative ways of saying it?
In American English, "The rain let up," is a very common idiomatic phrase.
In fact I think that expression might be just as commonly heard as, "The rain stopped."
"Let up" could mean that the rain completely stopped, or that it just became less intense, either momentarily or for the rest of the day. Or the expression could be used after a few rainy days in a row to indicate the beginning of dry weather for a day or more.