Items you receive automatically go in to your Backpack. You can see what is in your Backpack by pressing the B key when you're not in a duel. Remember B for Backpack!
Items that have the little hand icon are what you are currently wearing.
Items that have the little Crowns icon were purchased with Crowns.
Items that are greyed out, you cannot wear.
To wear an item, you can double click on it with your mouse, or select it and use the 'hand' icon at the bottom. If you receive an item as a reward, and your Backpack is full, the item will be put into your Bank.