Left Brain Right Brain would be better named Left Hand Right Hand, as the entire focus of the game is to help train your non-dominant hand to be as dexterous as your dominant hand (i.e. train you to be ambidextrous). If you can force yourself to sit through yet more minigames on the DS, ambidexterity may be an achievable (if not entirely fun) goal.
To get your non-dominant hand up to speed with your dominant hand, you play a series of minigame exercises that are focused on speed and dexterity. Most of these exercises are familiar (read: uninspired), including whack-a-mole, connect the dots, handwriting exercises, mazes, balloon popping, and so forth. There are a few more creative games, like Save the World, where you fling asteroids away from the screen with the stylus before they hit Earth.
There are five levels of difficulty, offering new minigame exercises and harder versions of previous minigames as you progress (there are 15 minigames total). Each exercise is always played with your non-dominant hand, and then once you've practiced enough you can test your weaker hand against your good hand, to see how closely they compare.
Your goal is to get your performance on each of the exercises to be equally strong for both hands. So, if you can pop 130 balloons in 30 seconds with your left hand, but only 95 with your right hand, you'll need to practice with your right hand more until you get the scores to match up.