To unlock all the achievements related to stars, you need to collect a total of 300 of them. There are 3 on each level and it is not that hard to collect most of them. Just play far enough into the game until you get to 300. You can check your total in the World (Box) Selection screen. You need to finish a level (by getting the candy in the creature's mouth, regardless of collecting any stars) to unlock the next level of the world. You unlock new worlds (the first level of each) by gathering a certain number of stars.
See THIS link for 3-star video walkthroughs for each level of the game if you are having trouble. Thanks to cuttheropecheats.com for posting these videos. Note that the worlds are numbered differently in the iOS version (which is what the videos are for), but the levels are all identical once you choose the correct worlds.