Live Weather is a feature that matches the weather in-game to a real and current weather report for that course. To be honest it is just a flashy way to show off the fact that rain has been implemented this time round.
Setting this up can be tricky. Firstly you need someone to play a full 18 hole match with you. And secondly, you have to find out which of the courses is currently getting rain. Before you set up the match, set up a single player round
by yourself and turn Live Weather on. Now when you go into the course select, you should be able to see the weather at that course in the info box on the right of the screen. Find one where it is bucketing down, and use that for your online match. Once the round is completed, you will receive this trophy.