Expansions are available at level 3.
Each specific expansion square has its' own requirements: a certain minimum number of Royal Points (RP), a certain number of Wisp Lanterns (L), and a certain amount of gold Coins ($). Beyond a certain area, Expansion Permits (P) are required. A special item (S) specific to a quest may also be needed. Certain expansions unlock new Characters, Buildings, and Quests.
The map below shows the unlock requirements for each square. It is a snapshot from September 03, 2021.
To expand, click on any "Expand" billboard in the forest, or click the Market menu in the lower right, which has an "Expand" option.
A map layout opens. Your existing kingdom map is shown, with additional transparent squares outlined in yellow outside your existing borders.
Click any expansion square to see the requirements and cost.
Unlike many menus, which show how much of a resource you have in your inventory versus what is required, this one initially shows a zero versus what is required.
If you are ready to commit resources to a specific expansion square, you can press "Use," and the committed number of resources appear instead of the zero. If you only have 2 lanterns and 6 are required, it will show 2/6.
WARNING: be aware that the resources become PERMANENT down payments on that square. You can't get them back to use elsewhere. The number of lanterns, coins, etc are no longer in your inventory. They are "attached" to that square, and will still be there the next time you check.
If you decide not to finish committing resources to the square, you can close the map (large "Done" checkmark at bottom right).
Alternatively, when all requirements are met, and payments are applied, you click to buy the square. It gets a big green checkmark on the expansion map. Click it, then close the map and enjoy using your new square.
Credit: map created and maintained by ShibuyaCloth
Reference: Official Forum Nugget #2 - Expanding