In the ume season, it rains a lot in Japan (and some other East Asian countries). That's why ume rain means rainy season. It usually happens from the end of May to mid-July, when ume get ready to be harvested.
Keep in mind this is a seasonal word unique to Japan and East Asia, so you usually don't use it to refer to rainy seasons that happen in other regions of the world.
This word uses an exceptional reading, so don't try to read this with the kanji readings you already learned. Here's a mnemonic to help you:
Rainy season is a special season to you (つゆ). All of your favorite things in the year happen during rainy season, like… staying inside all day because all it does is rain! But to you, there's nothing better than rainy season.
The rainy season lasts from June to July in Japan.
The rainy season started in Kanto yesterday.
I don't like the rainy season because it gets too humid, but I also don't like how it suddenly gets really hot once it ends.