Yes, this is a joke that plays on Ebonics. Snoop Dog is the name of an American rapper. "Fo'" is slang for "for" and drizzle means light rain.
So why does Snoop Dog wear a rain jacket? Fo' drizzle.
This joke may never truly be funny if you aren't a native speaker (it's not even that funny to begin with...) but hopefully now you understand it!
In Perfect English