Pimiento literally means "pepper". However, in this song, it's used in a very common and colorful Spanish idiom: "importar un pimiento".
This phrase is a colloquial way to say "to not care at all". When Manu Chao sings, "¿Será que a ti te importa un pimiento?", he's sarcastically asking his former lover, "Do you even care one bit?". Learning this song will teach you a fantastic, slightly cheeky expression that you'll hear in everyday Spanish conversation.
This song is a conversation between two people breaking up, and it's a total mess of conflicting feelings. Manu Chao kicks things off in Spanish with some seriously dramatic lines, like "Vivir sin ti es vivir muriendo" (Living without you is living while dying). He says living this way is so bad, "no lo recomiendo" (I don't recommend it). He feels like singing about her is like sticking thorns in himself, calling it his "droga ruina" (ruinous drug). It's all very intense, comparing her to powerful drugs and chasing after a shadow of her that he's invented.
Then Laeti comes in with the French response, and it's a firm, repeated "J'reviendrai jamais" (I'll never come back). She explains that even though there was love, she had to create distance: "Malgré le love, j'ai mis d'la distance" (Despite the love, I put distance). She has her own goals to follow, saying "Je dois exaucer mes rêves" (I have to make my dreams come true). She acknowledges the pain, singing that a song makes her cry, but she's clear that going back is not an option. It's a heartbreaking back-and-forth where he's spiraling in his addiction to her, and she's trying to move forward for her own sake.