AS far as I can tell, the goal is to get 10 points before anyone else. You have to eliminate 3 same color spheres at once to get points, I think. Rainbow colored ones can be substituted if you only have 2 color match.
You also have to make sure your homefield doesn't get covered up in all the appearing spheres while doing the chain eliminations. If it gets all covered up, you have a small time to clear them out beofre it automatically makes you lose the round.
But you probably already knew all that basic stuff. I can't provide anything deeper unfortuneatly. I don't understand Japanese either.