The Chest is a place to store your items. Items placed in a Chest can be taken out at any other Chest. They are located in the main towns. You can also build one when you get your own Island.
First, select an item in the Chest then use the buttons at the bottom of the screen to move the items about. The left side has "Withdraw" and "8" which allow you to take out one or eight selected items from the Chest and moves it or them into your Inventory. The right side has "all" and "Deposit" which takes all selected items from your Inventory, or one item at a time, moving them or it into your Chest.
The counters above the item represent how many are in the chest (upper left), or in your Inventory (upper right).
Enabling Mods enhances your options.