PĂ©pins literally means "seeds" (like from a fruit), but it's often used colloquially to mean "troubles" or "glitches." It's a charmingly understated way to refer to problems.
In this melancholic song, Aznavour uses "les pépins tristes compagnons" (the troubles, sad companions) to personify the difficulties that accompany the rain, creating a vivid and poetic image of sorrow. This double meaning makes it an interesting and memorable word.
"Il Pleut" ("It's Raining") turns a rainy day into a mirror for heartbreak. Aznavour describes an entire town going gloomy along with him: the gutters "have a drop in their nose" like they've caught a cold, the birds desert the sky, and even the windows look like they're crying.
Partway through, the weather turns explicitly personal. The narrator admits his heart is "en déroute," in retreat, because he's lost the person he loved. The rain outside becomes the tears he's holding in, and by the closing lines the two are fully merged: what falls from the sky is his own sorrow.
It's a classic chanson move: instead of naming the sadness directly, the whole physical world around him (houses, streets, sky) is drafted into feeling it too.