Desapeguei comes from desapegar-se, meaning to let go emotionally or detach yourself from someone or something. In the song, the singer admits, “Ainda não me desapeguei,” expressing that he still cannot move on from the relationship, making this an emotionally rich and useful word for learners.
The singer is still sleeping alone and hasn't adjusted to it: "Ainda não me acostumei / A ficar sem teu calor pra dividir meu cobertor" (I still haven't gotten used to being without your warmth to share my blanket). The person left the way a bird leaves, and the song keeps returning to that picture: "Você me levou pro céu depois voou" (You took me to the sky, then flew away).
He has love left over with nowhere to put it, and he asks "Me diz qual é a graça de um jardim sem flor / De um pássaro que não pode voar" (Tell me what's the point of a garden without flowers, of a bird that can't fly). His answer is small and a little childish on purpose: "Vou fazer de um papel, um avião / Pra te mandar essa canção de amor" (I'm going to make a plane out of paper to send you this love song). The paper airplane is the song itself, thrown after someone already out of reach.