À francesa literally means "in the French way" or "French style". It's an idiomatic expression that refers to leaving a place discreetly or without saying goodbye, often to avoid an awkward situation.
In the song, the narrator uses this phrase to describe how she left the table and the difficult conversation: "Caladinha à francesa" (Quietly, in the French way). This adds a touch of irony and sophistication to her exit from a rather dramatic breakup scene.
The whole song happens at one table. He brought her to the restaurant to confess, and the confession names the place twice over: "Sim, foi aqui que eu traí sua boca" (Yes, it was here that I cheated on you), "Foi aqui que eu fiquei com outra pessoa" (It was here that I hooked up with someone else). She works out what is coming before he gets it out, which is why the song opens on someone stalling for time: "Peraí, calma aí" (Wait, hold on).
What lands is how she ends it. No scene, no raised voice: "Hoje ela terminou comigo na boa, na classe e na educação" (Today she broke up with me nicely, with class and with manners) - the composure is the sting, and the song keeps returning to that line. He leaves "caladinho à francesa" (silently, a French exit) with "um gosto de raiva e um soluço de cerveja" (a taste of anger and a beer hiccup), and covers the last few metres "da mesa até o carro arrastando o coração" (from the table to the car dragging my heart).