Your Island Rank is the number you see in the top left corner when you're viewing your island. Increasing your rank will unlock new decorations and areas on your island.

To increase your Island Rank, you need Island Points. Earn Island Points by placing and upgrading decorations, as well as improving story buildings, on Mystic Bay. When you tap the house icon in the lower-left corner of the main island screen to access Decorations, you can see the number of Island Points that a decoration or building awards, displayed just below the item.

You can see how many Island Points you need to reach the next rank at the top-left corner of your main island screen.

Don't forget to upgrade your story buildings and decorations to receive the maximum Island Point value from them!