Me la suda is a colloquial Spanish expression, especially associated with Spain, meaning “I couldn’t care less” or “it makes no difference to me.” Literally, it translates roughly as “it makes me sweat,” but in the song “La fama me la suda” means that fame does not matter to him at all.
Álvaro de Luna sings about a man who is wrecking himself to avoid a heartbreak. Every night starts the same way: "Otra noche / Negando to' lo que hice ayer" (Another night / Denying everything I did yesterday), with car noise and a fog that "no me deja ver" (doesn't let me see). The chorus sounds like a boast, "Vivo como un rockstar / Las noches son de fiesta" (I live like a rockstar / The nights are for partying), but it ends on "Y ya perdí la cuenta / De las veces que me fallé" (And I've lost count / Of the times I let myself down).
The bridge drops the act. "No quiero / Ni puedo" (I don't want to / Nor can I), he says, then admits "Me duele pero debo olvidarte" (It hurts but I must forget you) and "Lo siento si no supe cuidarte" (I'm sorry if I didn't know how to take care of you). The party life turns out to be a way of running from her, and the last line explains it: "Es que intentaba salvarme" (It's that I was trying to save myself).