Bâillonne comes from the verb bâillonner, meaning to gag or muzzle. It's a strong, evocative word that isn't commonly found in everyday song lyrics.
In "Black Trombone," Gainsbourg uses it to describe how the music of the gramophone "gags" or "muzzles" his boredom (mon ennui). This personification of music silencing his ennui is a striking and memorable image, making the word particularly interesting and unique to this song's context.
"Black Trombone" is Gainsbourg's 1962 sketch of terminal boredom, half-spoken over a sparse jazz combo that mimics the mood it describes. The trombone becomes the emblem: "Black trombone, monotone" ["Black trombone, monotone"], one sluggish note standing in for a life stalled in place. A gramophone "bâillonne mon ennui" ["gags my boredom"] rather than curing it — even music can't drown out the tedium. A woman who "se donne à demi, nue, frissonne" ["gives herself halfway, naked, shivers"] breaks the gray for a moment, but it doesn't hold: "c'est l'automne de ma vie" ["it's autumn of my life"], and the song ends flat and resigned — "j'abandonne, c'est fini" ["I give up, it's over"].