Remontes comes from the verb remontar, which means to ascend, to soar, or to overcome. It's a less common verb than 'volar' (to fly) and carries a stronger sense of rising above challenges or difficulties.
In the song, it appears in the line, "Cuando vueles con tus alas y remontes las montañas" (When you fly with your wings and soar over the mountains). It beautifully conveys the idea of a child growing up, spreading their wings, and conquering life's obstacles, making it a very evocative and inspiring word.
Carlos Vives wrote this one as a father talking straight to his kids, and he opens with a plan rather than a feeling: "Mañana nos vamos a la playa / A nuestra casita en el mar" (Tomorrow we are going to the beach, to our little house by the sea). Then he starts counting what he got wrong. "Las noches que no estuve en Bogotá" (The nights I was not in Bogotá), and "Y no te persigné antes de dormir" (And I didn't make the sign of the cross on you before you slept). It is a strange, tender bit of arithmetic, adding up the bedtimes he missed while he was away working.
The song's real move is who gets left behind. The children fly off, "Cuando vueles con tus alas y remontes las montañas" (When you fly with your wings and soar over the mountains), and the father stays: "Y yo me quedo a sobrevivir / Al silencio de la casa / Al balón en la terraza" (And I'm left to survive the silence of the house, the ball on the terrace). That ball, sitting where nobody is playing with it, does more work than any line about missing them could. The chorus he keeps returning to is the promise underneath all of it, "Nunca olvides que yo siempre estoy aquí" (Never forget that I am always here), and by the last verse the beach house is full again, with the cousins, the grandmother, Pete, Elena and the whole in-law side turning up to say hello.