Abrigo literally means "coat" or "shelter." However, in this song, it's used metaphorically to describe finding comfort and security in a new relationship.
The lyrics say, "Y saber que ahora tú ya tienes abrigo" (And to know that now you already have shelter/warmth), implying that the person the singer loves has moved on and found solace with someone else. This makes the word particularly poignant and interesting.
"Se Va Muriendo Mi Alma" ("My Soul Is Slowly Dying") is a banda ballad about the ache of being left behind. The narrator watches the woman he loved move on to someone new, and describes his own heartbreak as something physical: his soul dying a little more each day, an emptiness growing inside him that nothing can fill.
He keeps circling back to what he can't let go of: this was the one time he really knew how to love, and now he has to give it up. The hurt lingers because her love never fully left him, it just stopped being returned. The cruelest part is that she doesn't even remember him anymore, while he can't stop thinking about her.
It's a song about one-sided heartbreak, holding on to a love that has already ended for the other person.