You can claim the cards by opening the Chests you've collected. The number of collected Chests is displayed on the "Chest" box at the bottom left corner inside the "Collection" button. To open a Chest, tap on the "Chest" box, select the Chest you want to open, and tap on the "Claim" button.

If the collected card is unique, it'll automatically appear in the corresponding collection book. New cards are marked as "New" once they appear after opening the Chest.