Grelottant means "shivering" or "trembling from cold". It's a vivid and evocative word that paints a clear picture of the physical sensation.
In the song, it's used to describe the protagonists sleeping in their coats and "grelottant" in their attic rooms, highlighting the youthful, somewhat impoverished, yet carefree existence they once led. It's a less common word than simply "froid" (cold), making it more interesting and memorable.
Joe Dassin's "Le Temps Des Oeufs Au Plat" is a nostalgic look back at a friendship and the passage of time. It starts with a vivid picture of youthful, carefree days: "Toi, tu portais la barbe et moi / J'avais des bottes qui prenaient l'eau" [You, you wore a beard and I / Had boots that leaked]. They'd play Brassens songs badly on guitar and banjo, and instead of going home, they'd pretend to flirt or just talk for hours in cafes. It was a time of simple pleasures, like "le temps des œufs au plat" [the time of fried eggs], and sleeping in cold attic rooms, only warmed "quand une fille nous tenait chaud" [when a girl kept us warm].
Fast forward, and things have changed. Marriage, which they once saw as "le service militaire à vie" [military service for life], has come. The singer is now a witness, and his friend the groom. Their future meetups will be planned by agenda, with polite gifts for the wife. The wild days are replaced by "le temps des dimanches au bois" [the time of Sundays in the woods], with folding tables and cold chicken. But even as they become respectable, there's a hint that the old spirit isn't entirely gone. When they're older, perhaps after their wives have gone to do the dishes, they'll find themselves reminiscing over cognac and decaf coffee, returning to "le temps des œufs au plat" [the time of fried eggs] and their good old days.