Desvelado describes the state of being sleepless or having stayed up all night. It's a very expressive word that goes beyond simply being 'awake' (despierto).
In this classic song, the singer is desvelado because he's wandering the streets, obsessing over a woman's voice he heard on the radio. The word perfectly captures that feeling of lovesick restlessness and longing that is literally keeping him from sleeping.
Have you ever fallen head over heels for someone you've never even met? In "Desvelado", Tejano singer Bobby Pulido tells the story of a guy who falls in love with a voice he hears on the radio. He admits right away, "Con ella me enamoré / Que nunca la conocí" (With her I fell in love / Though I never met her). It is dramatic, a little obsessive, and completely catchy. Instead of sleeping, he spends his nights wandering around, tuning in every single day just hoping to hear that tender voice again.
The title word itself gives you the whole mood: desvelado means sleep-deprived or kept awake by thoughts. He wanders the streets wondering "¿Dónde andará?" (Where could she be?), totally consumed by longing. He even looks up at the sky, singing "Por el cielo busco mi estrella / A la luna quiero subir" (In the sky I look for my star / I want to climb to the moon). Even though waking up without her makes him want to cry, he refuses to give up the search. He is committed to staying "desvelado y sin amor" (sleepless and without love) until he finds the angel behind the microphone.