Resource Chests are chests that can be found whilst exploring the map of which there are many. Some are in plain sight with others hidden from view which you have to excavate out of mountains, grassland and the ocean. These are highlighted by confetti floating around the area they are hidden in. During the night once Calendar are researched they are easier to see as they glow in the dark.
All of them provide Resource Cards and come in variety of designs although presumably they will provie Stickers after the next Update. Once they have been opened they shrink and disappear from the world and all of the chest (not just the lid) have to be excavated to be opened. If the player has already maxed out all his available resource cards, the chest will give instead a Copper Idol Card which provides +1000 belief.
To dig down through layers to access chests simply double click with the right mouse button where you wish a hole to appear. Shift-D and Shift-A will rotate the camera, allowing you to search on the sides of mountains easier.