Enclume translates to "anvil". It's an interesting and somewhat rare word to find in song lyrics, especially in a context describing sleep.
In the song, the narrator describes his love sleeping "du sommeil de l'enclume" (with the sleep of the anvil). This vivid metaphor suggests a very deep, heavy, and undisturbed sleep, much like the solid, unmoving nature of an anvil. It's a creative and memorable image that stands out.
Renaud's "La Pêche À La Ligne" is a charming and humorous look at a morning in the life of a man who'd rather be fishing than dealing with domesticity. He wakes up before his partner, who's still in a deep sleep, dreaming of Marlon Brando – a detail he finds quite funny, thinking, "Qu'est-ce que je foutrais là?" [What would I be doing there?]. While she's off in her movie-star dreams, he's sharpening his fishing poles, packing a "Saucisson, bière fraîche" [sausage, cold beer] for his trip.
As the day goes on, his partner wakes up, makes calls to her mother, and puts on her makeup. He wonders who she's trying to impress, musing, "Je me tue à lui dire / Qu'elle est mieux sans tout ça" [I kill myself telling her / That she's better without all that]. Meanwhile, he's out by the water, having a bit of a clumsy time, even hurting his finger. He comically catches a "truite / D'au moins cent vingt kilos" [trout / at least one hundred twenty kilos] but throws it back because it's "trop petite" [too small]. He returns home empty-handed, only to find his love has left him. Despite the heartbreak, he plans to take his daughter fishing next Sunday, finding a small comfort in the simple act of "s'emmêler dans les branches / À la pêche à la ligne" [getting tangled in the branches / while fishing].