Asómate comes from the reflexive verb asomarse, which means to lean out, peek out, or show oneself at an opening like a window or door.
In the song, Romeo Santos begs his love, "Te lo ruego, asómate a tu balcón" (I beg you, lean out your balcony). Calling himself a "modern Romeo," he uses this highly visual verb to recreate the classic Romeo and Juliet balcony scene, perfectly setting the stage for his desperate and romantic serenade.
Romeo Santos shows up under a balcony barefoot, drunk, and dramatic, begging his love interest to look out her window. He claims he has already written three bachata tracks to prove his feelings, dubbing himself a crazy bohemian and "tu Romeo moderno" (your modern Romeo). Even if the police come to drag him away, he just asks for one goodbye kiss before proposing the wildest romantic wager: give him exactly one week, and he will make her his wife.
From there, he lays out an ambitious day-by-day battle plan across English and Spanish. Sunday starts the game of seduction, Tuesday is a trip to become an official couple in Paris ("Martes te hago mi novia en París"), Thursday brings the ring on bended knee, and by Saturday, they are off on their honeymoon. Calling himself "el chico de las poesías" (the poetry guy), he insists that love does not care how long you have known someone, promising the complete fairy tale in a neat seven-day package.