Vuélveme loco literally translates to "turn me crazy" or "make me crazy." It's an idiomatic expression that means to drive someone crazy, often in an intense, passionate, or even desperate way.
In the context of "Sin Despedida," Carlos Rivera uses this phrase to express a profound plea. He's asking his lover to make his departure so overwhelming and disorienting that he won't even realize she's leaving. It's a dramatic and emotional request to avoid the pain of a direct goodbye, making it a very memorable and impactful phrase in the song.
Carlos Rivera and Alejandro Fernández spend this whole song asking for the one thing nobody ever grants: a breakup with no conversation. It opens already resigned. "Sé lo que vas a decir" (I know what you're going to say), "Porque hace tiempo no te veo feliz" (because I haven't seen you happy in a while). So the plea that follows is not for her to stay. It is for her to skip the speech: "No me digas nada, que es mejor así" (Don't tell me anything, it's better this way), then "Da la media vuelta, que yo haré de cuenta / Que nunca llegamos aquí" (Turn around, I'll pretend we never got here). The title line puts the request plainly: "Haz tu partida / Sin despedida" (Make your exit, without a goodbye).
The chorus is worth staying for because the demands keep escalating. "Vuélveme loco" (Drive me crazy). "Déjame ciego" (Leave me blind). "Miénteme un poco" (Lie to me a little). Each one asks for a stronger anaesthetic than the last, until he is asking to be finished off outright: "O quítame, vida / La misma vida / Y déjame muerto" (Or take from me, my love, life itself, and leave me dead). It is a lot, and then the closing line explains why he thinks none of it would cost him anything. "Que yo ya estoy muerto" (Because I'm already dead), "Por dentro" (on the inside).