These achievements will all be obtained along the path to collecting the 231 crystals in the game. There is an achievement for 50, 100, and 200 crystals, as well as three for collecting all the crystals in each of the game's three planets.
The puzzles in the game are not that difficult and you should have little trouble gathering all the crystals and making it to the spaceship. The number of crystals in each level varies from 3 to 5. You can tell which levels you are missing crystals on because they will appear greyed-out in the level-select screen. The ones you have already collected appear red.
The "X planet explorer" achievements do have somewhat misleading descriptions. They are not awarded for just completing the levels (which you do by getting the alien to the spaceship). They are actually for collecting every available crystal in that world.