Within Sunshine Days, an Island is an area where you are able to gather resources, unlock new NPCs as friends and interact with unique harvestables, themed by the island they're located in. These resources will be needed for filling order boxes, for completing certain tasks and crafting items.
To be able to reap the maximum amount of resource from an Island, you will need to help out the Island's unique NPC, for example, Rowan in the Lazy Lumberyard.
Each island will usually introduce a new method of obtaining a resource. For example, chopping trees in Lazy Lumberyard for logs or mowing through flax in Scarecrow Meadow. After a resource is collected, it will have a cooldown that will need to finish before you can harvest that resource point again.