“When's this rain gonna let up?”
It's been raining a lot for the last few days. You look outside and it's still raining hard today. You comment on it to your roommate.
When's this rain gonna let up?
When there's weather that's annoying or uncomfortable, you want it to stop. In this case, you can use the phrase "let up":
When will this heat let up? My God, it's scorching!
When a person is nagging or criticizing you, you can ask them to "let up":
A: How many times do I have to ask you to throw away your empty bottles after you're done with them?
B: Will you let up? I'm doing something important here!
"Gonna" is a way of pronouncing "going to". You use it like this:
What're you gonna do?
Native English speakers sometimes mash these two words together. It actually sounds more like "gunna", but "gonna" has become the accepted way of spelling this when you want to capture the sound of someone's ungrammatical speech.