The TheyMightBePowers are a form of elemental creative energy associated with They Might Be Giants. They give their wielders the power to create or channel phenomena associated with one or more of TMBG's songs. They are administered by a cosmic being known as Size-of-the-Entire-Universe Man.
Size-of-the-Entire-Universe Man bestowed the TheyMightBePowers on Person Man by hitting him with a cosmically charged frying pan from space, transforming him into Particle Man. The frying pan shattered into many pieces which gave the powers in a more limited form to others: a group of bullies who were attacking Person, who became Pencil Rain, as well as Ana Ng.
Project Serra's plot to channel extradimensional energy through They Might Be Villains Mountain in Birds Fly seemed to have something to do with the TheyMightBePowers, suggesting They Might Be Villains drew on them too even though they weren't from the Looniverse.
Occultism Kid speculated the TheyMightBePowers were a form of "universal magic" inherent in the Looniverse itself. Like the Power Kirby, they may also be a manifestation of the Source Code channeled through particular artists, which would explain how the undead James K. Polk was able to wield it in spite of having been born 200 years before the band existed.