BGM refers to music as the background, which is often used to render the atmosphere.
Bring your own BGM means that every time a character appears, BGM will switch to the music exclusive to the character, or the music conforming to the character temperament, or other specific music. It is often used to describe a person who has a strong aura, or to express that this person is a typical person. (The more "correct" explanation is that BGM does not change with the appearance of characters, but the characters bring their own equipment to play BGM. )
Wu Ke's BGM probably comes from the crazy father of RPG horror game, which has a mysterious bald demon character. As an important supporting role and clue of the game, BGM suddenly changes every time it appears, and is described by players as a man with BGM.