Presiento comes from the verb presentir, meaning "to have a premonition" or "to sense something beforehand." It's a more evocative word than simply "I think" or "I feel."
In the song, Alex Ubago sings, "Aunque presiento que ya es tarde para esta canción" (Although I have a premonition that it's already too late for this song). He uses it to express a deep, intuitive feeling about the futility of his apology, adding a layer of emotional depth and resignation to his plea for a second chance.
Have you ever messed up so badly in a relationship that you just desperately want another chance? That's exactly what Alex Ubago is singing about in "Te Pido Otra Oportunidad" [I Ask You For Another Opportunity]. He's really laying it all out there, admitting, "Qué hice mal" [What did I do wrong?] and acknowledging that sometimes he loses his temper.
He's not trying to forget his ex, saying, "Quise olvidarte pero no" [I wanted to forget you but no], because she's a part of his life. He's promising to make things right, to heal the wounds, and even promises, "Yo te prometo que no vuelves a llorar" [I promise you won't cry again]. It's a really heartfelt plea for a fresh start, asking her to look him in the eyes and see that he's sincere. He just wants to fix what he broke!