Madrugada refers to the very early hours of the morning, specifically from midnight until dawn. It's a more evocative and specific term than just saying "morning."
In the song, the artist sings, "Porque en la madrugada si no siento tu piel" (Because in the early morning if I don't feel your skin). This word sets a melancholic and intimate scene, highlighting the loneliness and longing he feels during the quietest and often most vulnerable part of the night without his loved one.
Gerardo Ortiz's "Historia de Ayer" is a heartfelt plea from someone who's been left behind and is really struggling to cope. He's desperate for an answer, asking, "Quiero que me digas si vas a volver" [I want you to tell me if you're coming back] because he truly doesn't feel good without his love. It's clear he's hurting deeply, especially in the early hours, when he misses her touch and looks at her picture, realizing "lo más duro de esto es que no sé que hacer" [the hardest part of this is that I don't know what to do].
He's trying to understand how she could have left him, and the pain of realizing "que solo fui tu juego y me tocó perder" [that I was just your game and I had to lose] is hitting him hard. The song really gets to the core of that crushing feeling when you finally accept that it's over, and you're no longer anything to that person – just "solo historia de ayer" [only history of yesterday].