Duérmete is the informal command form of "dormirse," meaning "to fall asleep." While seemingly simple, its usage here is quite striking and unexpected.
In the lyrics, the phrase "Duérmete tan perra me tocó" is a colloquial and somewhat harsh way of saying "I got such a lazy/worthless woman." The direct command to "fall asleep" in this context implies a deep disappointment and perhaps a wish for the person to simply disappear or stop being a problem. It's an emotionally charged word that reveals the singer's frustration.
This song tells a classic story of chasing a dream and getting a rude awakening! Our singer heads north with a big dream, or "grande ilusión," to make his life better and help his family. He's sending money back home, a lot of it, thinking he's doing right by his love.
But oh boy, does he get a shock when he returns! He finds out his partner was a "perra" — a harsh word, meaning something like a dog or a bitch — and totally betrayed him. She didn't care for him at all, and now he's left heartbroken, drinking to forget her because the memory of her isn't easy to shake. It's a really relatable tale of love lost and the pain of a dream turning into a "falsa ilusión" [false illusion].