Veste literally means "jacket". However, in French, the phrase "retourner sa veste" is an idiomatic expression.
It means to "turn one's coat" or "change one's mind/allegiance", often implying a betrayal or a shift in loyalty for personal gain. In the song, the singer uses it to accuse her partner of changing his feelings and betraying their love, making it a very impactful and culturally interesting word choice.
Amel Bent's "Pourquoi Tu Restes" [Why Are You Staying?] is a powerful and direct question to a partner who seems to have checked out of the relationship. The singer is clearly hurt, asking, "Pourquoi tu restes? T'as bien retourné ta veste" [Why are you staying? You've really turned your coat], implying a betrayal or a change of heart. It's a blunt observation that the love is gone, with lines like "Tu m'dis même plus 'je t'aime'" [You don't even tell me 'I love you'] and "Ça s'voit que tu me détestes" [It's clear you hate me].
She dismisses common reasons for staying, like promises that are now "loin tes promesses" [far away your promises], or even for the kids, saying, "T'inquiète, ils sont déjà grands" [Don't worry, they're already grown]. The singer feels the pretense is painful, "Tu vois bien que ça me blesse" [You see that it hurts me], and knows "c'est la fin du jeu" [it's the end of the game]. It's a raw and honest look at a relationship that's clearly over, even if one person is still physically present.