Collecting Pages is one of the main markers of progression in New Super Lucky's Tale. (They're sorta like Jiggies, if you're familiar with Banjo-Kazooie). Each level has 4 pages available for you to collect. You earn one page for completing the level, one page for collecting at least 300 coins in the level, one page for finding all five of the letters "L" "U" "C" K" and "Y", and one page for completing an additional hidden task within the level. You can replay a level to go back and earn any pages you missed, and you don't have to earn all four pages in one run of the level.
To teleport to Sky Castle, pause the game from any world, select the second option "World Select," and then "Sky Castle." There are four levels in Sky Castle: Heavy Heads, Aqueducts, Go Fetch, and Dungeon Run. Each of these levels have the standard four pages to collect. If you're having trouble finding the letters or the hidden task in any level, I've found this very useful text/picture guide which has all of the letters and hidden task locations: https://www.thegamer.com/new-super-luckys-tale-sky-castle-hi...
The last six pages in Sky Castle come from foxhole puzzles. Dive into the foxholes around Sky Castle with
. In each of these foxholes you will be faced with a basic sliding puzzle. None of them should be any tricky at all, but if you're having trouble, the above link has solutions to each of the sliding puzzles as well. Once you complete a foxhole, a green clover will appear on the foxhole to let you know that you've already collected the page from that foxhole. Foxholes with empty clover outlines have not yet been completed.
Once you collect all 22 pages in Sky Castle, this achievement will unlock. Enjoy New Super Lucky's Tale!