Crystals allow you to buy a Crystal Barrel in the Store. Crystal Barrels contain Cards and Cash!
Earn Crystals by selling, or burning, Cards you don't need.
Remember that card upgrades are in multiples of 5, so it's a good idea to keep round numbers of cards while selling the rest. For example, if you have 27 cards for Kitten Mittens, and it's level 2, that means level 3 will cost 10 cards so you may want to keep 10 cards and sell 17.
Upgrade Dee to increase your Crystal Power, which increases the number of Crystals you earn per card burned so you get more bang for your excess cards.