Despegaste comes from the verb despegar, which literally means "to unglue" or "to detach." It's often used when something takes off, like an airplane.
In this song, Laura Pausini uses it beautifully and metaphorically: "Despegaste del cemento mis zapatos" (You unglued my shoes from the cement). This imagery conveys a sense of liberation and freedom, as if her lover helped her break free from being stuck or grounded, allowing them to "fly away for a while."
Laura Pausini takes Shakira's 1995 song and sings it straight, as a goodbye letter that keeps getting distracted by small things. Instead of listing what went wrong, she lists what the other person taught her, and most of it is oddly domestic: "Junto a ti, creo que aumenté más de tres kilos" (With you, I think I gained more than three kilos), "Y fue por ti que aprendí a querer los gatos" (And it was because of you that I learned to love cats), "Desarrollaste mi sentido del olfato" (You developed my sense of smell). Gaining weight from too many kisses is a funny thing to put in a breakup song, and that is exactly why it works.
Then the joke closes. "Pero olvidaste una final instrucción" (But you forgot one final instruction), and the missing instruction is the only one she actually needed: "Porque aún no sé cómo vivir sin tu amor" (Because I still don't know how to live without your love). The title means anthology, a collected edition, and the song is built like one. Every small thing the love left behind gets written down, item by item, until the last four lines just repeat "Lo que es amar" (What it is to love), as if she has run out of other ways to say it.