Desvelándome comes from the verb desvelar, meaning "to keep awake." The -me ending makes it reflexive, so it translates to "keeping myself awake" or more commonly, "losing sleep."
In the song, Ivan Cornejo sings, "Yo desvelándome pa' sólo ayudarte" (Me losing sleep just to help you). This single word powerfully illustrates the personal sacrifice and one-sided effort in the relationship he's mourning. It paints a vivid, relatable picture of staying up all night for someone who ultimately broke his heart.
This song is a straight-up anthem for a broken heart. The title itself, "Corazón Frio," means "Cold Heart," and Ivan Cornejo gets right to the point in the first lines: "Mi corazón, ya está frío / Me lo quebraste ahora no creo en amor" [My heart is already cold / You broke it, now I don't believe in love]. He's not just sad; he's completely disillusioned with love. His only way to deal with the pain and the memories of a relationship that he says "era serio" [was serious] is through his music. He tells us that his guitar is his outlet, because "Cantando es mi única forma de expresar" [Singing is my only way to express] his feelings.
He really goes into the damage this relationship caused, claiming it affected his physical well-being: "Y mi salud, me la dañaste" [And my health, you damaged it]. He accuses his ex of wasting his time and lying, saying all the times she said she missed him were just "falsas palabras" [false words]. In the end, he's left with the pain and the realization that while he wanted to be with her, she is now "nada" [nothing] to him. It's a blunt and emotional look at how a bad breakup can leave you feeling empty and cold.