버리다 here is like 'out or off' in some English phrases. It doesn't carry any big meaning.
Maybe it adds more emotion as regrets, sadness, or worries.
나 숙제를 잊어버렸어. I forgot my homework.
=나 숙제를 잊었어. (not normal to say this)
난 버스를 놓쳐 버렸어. I missed the bus.
=난 버스를 놓쳤어.
이 문제를 틀려버렸어. I got this question wrong.
=이 문제를 틀렸어.