Astío translates to "weariness," "boredom," or "ennui." It's a more sophisticated and less common word than simply saying you're tired or bored, carrying a deeper sense of emotional exhaustion or profound disinterest.
In "Da La Vuelta," Marc Anthony uses it to describe how love has escaped "por el portal del astío" (through the portal of weariness). This personifies astío as a gateway through which love can depart, suggesting that the relationship has withered due to a profound sense of emotional fatigue or indifference, rather than a dramatic conflict. It's a poignant and evocative choice that adds depth to the song's narrative of a fading relationship.
A breakup song that spends its first half asking questions and its second half refusing to beg. Marc Anthony works through everything she has apparently stopped feeling, then lands on the line that stings: "Que no basta con la piel / Para llenar el vació" (that skin isn't enough to fill the void). The chorus is the exit. "Da la vuelta y vete ya" (turn around and leave now), "Hoy te doy la libertad" (today I give you your freedom). He is holding the door open for her.
Then he admits what it costs him, once, and moves straight past it: "Y mientras tanto mi amor / Yo guardaré mi tristeza" (and meanwhile, my love, I will keep my sadness). The last third is where the song gets good, because the band opens up into a salsa montuno and the wounded politeness drops away. "Si no me quieres / Tú te lo pierdes" (if you don't love me, you're the one missing out) is pure bravado over a horn section, and he even shouts a cue to his own percussionist: "¡Opa! ¡ataca Cuko!" (Opa! Take it away, Cuko!). By the fade he is repeating "Vete que poquito a poco yo te olvidaré" (go, because little by little I'll forget you), which sounds less like a fact than like something he is practising.