You'll unlock this achievement after you earn 120 stars and complete the short sequence after returning to the ship. It takes 20 stars to unlock each 'zone', so you'll need to earn an average of 20/30 stars per zone, or 2/3 stars per level.
Play Kat's game in the arcade to get hint tokens if you need them. They show you how to play each level to get 3 stars. Then it's just a matter of doing it.
(Note that once you win so many hints, you have to start doing 2 robots at a time, possibly more later.)