Arrepiento comes from the reflexive verb arrepentirse, meaning 'to regret'. It's a powerful word used to express deep remorse for a past action or inaction.
In the song, David Bisbal sings, "Te lo juro que ahora me arrepiento" (I swear that now I regret it). He is filled with regret for not saying "I love you" in time, and this intense feeling is the driving force behind his plea for his love to come back.
This song is a huge, dramatic apology from someone who made a very big mistake. The singers admit right away that they broke a promise: "Nunca fue tan falsa una promesa... Cuando te juré que ya no pensaría en ti" (A promise was never so false... as when I swore I wouldn't think of you anymore). They tried to move on, but it was a total failure. They even lied to their ex's friend, saying they were over it, but then confess to us: "Fue mentira... Porque sí te extraño" (It was a lie... Because I do miss you).
They are completely convinced that their ex is going to come back, saying "Vas a quedarte sin pretextos para huir" (You're going to run out of excuses to run away). The whole song is basically a desperate plea for a second chance. They beg their ex to "vuelve, vuelve" (come back, come back) because without them, they're a "desastre" (disaster). It's all about that feeling of deep regret when you realize you let the right person go and you'll do anything to get them back.