You have to reach the exit door to pass to the next level. In the levels where they are keys you have to collect them to open the exit door. Many times you'll have to invert colors and positions vertically in the game to be able to go on. And hurry up, because the time is limited!
Arrow keys to move to left/right; use the P key to see the menu. Space bar to jump. SHIFT key to invert vertically the game (make the ground to be roof and conversely, the white to be black and conversely, etc.). This inversion is key to be able to progress in the more difficult levels.