To "get caught" in the rain means that you are outside and it starts raining suddenly, before you can get inside or get an umbrella.

The phrase "get caught" is interesting. It's connected to the phrase "catch someone":

A police officer chased the thief down and caught him.

From that meaning came a secondary meaning, which means to see someone doing something that they wanted to hide or keep secret:

I caught Ed watching YouTube videos in his office the other day.

Tiger Woods got caught cheating on his wife with a restaurant hostess.

Another meaning came from that which is to be suddenly put in a situation that you're not prepared for:

Neal got caught unprepared by the interviewer's question.

This is the sense that "get caught in the rain" is closest to. It means that you are unprepared for the rain when it starts.