The Collectibles are found in chests! The Collectible cards give you bonuses in your mines and different outfits for your workers.
Chests come in 4 different rarities: Common, Rare, Epic and Legendary. You have a higher chance of getting cards which match the rarity of the chest itself. You can get chests from Expeditions, from entering the game, and collecting a free wooden chest every four hours from the in-game shop.
You activate the bonus on a Collectible card by clicking on the card you wish to use, and then "Activate". You can change your chosen card at any time,and collecting more of the same card will eventually level it up so you can get more of a bonus.
You can view your cards and your unopened chests by clicking on the Collectibles island on the world map.