As described in Chapter 10 of the Blood Lad manga, there is a miasma located in the atmosphere of the demon world. This miasma, called Mazo, is where all magical abilities are obtained from. By absorbing it into the body, demons are able to gain their power. Essentially, a demon's body converts Mazo (the miasma) into their Maryoku (the individual's magic), which they expend by using their Maryoku (offensively or however their particular Maryoku may be used). When a demon dies, their Maryoku is sent to the Human world, as a soul, which is reborn into a Human. When Humans die, their soul travels to Makai in the form of Mazo, which normally would disperse naturally in the air. However, due to one King's (Grimm) greed and ambition of building a Door to control the flow of the Mazo, it is instead simply dammed behind the Door.
Fuyumi Yanagi is the exception to this, due to her being a ghost who can potentially obtain various kinds of magic.