Beat the first world to open Expert World, then collect stars to gain new levels.

Level X-1 - Get 9 stars
Level X-10 - Get 90 stars
Level X-11 & 12 - Beat the first 10 worlds with all presents
Level X-2 - Get 18 stars
Level X-3 - Get 27 stars
Level X-4 - Get 36 stars
Level X-5 - Get 45 stars
Level X-6 - Get 54 stars
Level X-7 - Get 63 stars
Level X-8 - Get 72 stars
Level X-9 - Get 81 stars