Stars are earned by replaying Hidden Object Scenes. The final Adventure Scene in each chapter requires a certain number of Stars before you can unlock it. Sometimes this means you have to go back and play scenes you have already played, to earn the extra Stars you need to progress. Higher scores in the Hidden Object Scene will earn Stars faster.
Sometimes a Star can be very close to completion but not quite, and playing the scene just one more time will earn you that Star. Remember, each time you earn a Star you will be awarded a Star Box, and the last Star in each scene will award you a rare 5-Star Box. When all five Stars have been earned, the scene is labeled with a red "Mastery" ribbon.