Recapacitar means 'to reconsider' or 'to reflect'. It's a more formal and introspective word than simply 'to think' or 'to change your mind', implying a deeper process of thought and realization.
In the song, the singer uses it when he says, "Pero a tiempo logré recapacitar" (But in time I managed to reconsider/reflect). He realizes he almost lost his love and takes the time to reflect on his actions, showing a moment of personal growth and a turning point in the narrative of the song.
Diomedes Díaz spends this whole song asking for one thing and never gets around to asking for anything else. He opens with a condition, "Ay, yo quiero ser feliz / Pero cerca de ti" (Oh, I want to be happy, but close to you), and then says what happens if the condition fails: "Yo me muero" (I die). The chorus is that same sentence again with the accordion behind it, and Juancho Rois answers him back on it between every verse.
The middle is where he admits fault. "Las cosas se pierden si uno no las cuida" (Things get lost if you don't take care of them) sounds like something an uncle would say over lunch, and then he turns it on himself: "Yo casi te pierdo por poco mi vida" (I almost lost you, my love). He never promises to stop being the kind of man who nearly loses her. He asks to be forgiven for it, "Si soy humano y cometo errores" (If I'm human and I make mistakes), and then asks her to put the memory somewhere she cannot reach, "Por favor aparta de ti esos recuerdos" (Please take those memories away from you). That is a lot of pleading for four minutes, which is the point.