If you have a passion for art, and making it is all you can think about or do, your passion for art is consuming. It is intense and almost overwhelming.
Picture a fire burning in a fireplace. The fire consumes the logs you have put in it. It completely devours them in order to burn. When something, a passion or a drive, is consuming, it dominates a person's life like that fire, and in a sense, it devours their energy and attention. Your consuming desire to travel might cause you to study abroad for a semester or even move to Europe.