To add a bit, it comes from 후비다 (pick, scrape, dig) + 내다 (take out, pull out). So it means pick or gouge something out of a small space or enclosure, like nose, ear, etc. It is also used figuratively to mean "gouging a hole out of one's heart", like "break one's heart".

Korean people are usually using this verb with 2 situation. 1) As digging something. ex) 코를 후비다. 귀를 후비다. 2) As finding emotional thing. ex) 이 문장은 내 가슴을 후벼판다. 저 노래는 내 가슴을 후벼판다.