Desdoblarme literally means “to unfold myself,” but it can also suggest splitting or duplicating oneself. In the song, the narrator imagines separating from his own perspective so he can look at the olive tree, and himself, from the outside, making this a striking image of self-reflection and escape.
Olivos is a sad song with a soft landing. It opens on a strange coping trick: "A veces me imagino muerto" (Sometimes I imagine myself dead), and that thought "me ayuda a relajarme" (helps me relax). Then the singer turns into an olive tree gone dry, "Como un olivo que se parte / Bajo sus propias ramas" (Like an olive tree that splits / Under its own branches). That is a sharp picture of being crushed by your own weight, and olive trees are everywhere in Spain, so it lands close to home.
The chorus is where it opens up. He wishes he could step outside himself, "desdoblarme" (split myself in two), and talk to that tree from the outside. What he would tell it is about a place where the air "Huele a sal" (Smells of salt), and a promise: "Que a todos los olivos miserables / Vendrá la primavera a abrazarles" (That all the miserable olive trees / Spring will come to hug them). By the end the pronoun shifts from "abrazarles" (to hug them) to "abrazarnos" (to hug us), and he finally counts himself in. Small grammar change, big feeling.