Barré comes from the verb se barrer, which is a colloquial French expression meaning "to leave quickly" or "to bolt". It's more informal and impactful than simply saying "partir" (to leave).
In the song, Tayc sings, "Si j'me suis barré c'est d'ta faute" (If I left, it's your fault). This word choice emphasizes the abruptness and perhaps even the emotional charge behind his departure, making it a more vivid and engaging term for learners.
Tayc spends this entire song failing to talk himself into a new relationship. He opens with the arithmetic of a breakup — "Ça fait des mois qu'on a refait nos lifes" [It's been months since we rebuilt our lives] — and by the second verse he has already moved on with someone else: "J'me suis mis avec une autre" [I got with another woman]. The trouble is that he keeps grading her against the woman he left. The hook is the verdict he keeps returning to, "Elle ne sait rien faire comme toi" [She can't do anything the way you do], and he repeats it until it stops sounding like a complaint about her and starts sounding like one about himself.
What lifts it above ordinary regret is that Tayc knows exactly how unfair he is being. His friends tell him the obvious thing — "ne compare pas la femme que tu as avec la femme que t'avais avant" [don't compare the woman you have with the woman you had before] — and one of them uses his name to slow him down: "Tayc n'fais pas ça, s'te plaît prend l'temps" [Tayc, don't do this, please take your time]. He hears the advice and ignores it. He even concedes the split was partly his own doing while asking to come back anyway: "Tu m'as fait du sale mais j'donnerai tout pour rentrer" [You did me dirty, but I'd give everything to come home]. By the final minute the words have run out and he is left humming the melody over the beat, which is the most honest stretch of the song.