After the round you will have to play 2 more rounds to finish the tournament, if you had a lower score, restart the Case, if you had a higher score, you will win and unlock. You get more points based on how many of your balls were closer to the white ball than your opponents. After you throw one, it will be your opponents turn and you continue back and forth until you both throw all 4. You will want to throw the Red Balls in the same manner (making sure to throw at half power/middle strength and aim the curved marker at the white ball. You will have 4 Red Balls and your opponent will have 4 Brown Balls. When you throw, press once to start up the power meter, then press again when the meter is around the middle to throw it out onto the lawn.
The goal of Boules is to start by throwing the white ball out onto the lawn. See the Video Walkthrough for a visual example. This is the beginning of the missable Skill based Achievement I mentioned before, so get ready for a fun time or a difficult time.