다가 보다

doing it a while, you notice

Two readings, both present here. The plain one is seeing something while doing something else: 알바 마무리하다가 봤어요 "I watched it while closing up my shift", 넘기다가 본 방법 "a method I came across while scrolling". The more important one is 다 보니까 / 다 보면 — "as you keep at it, you find that…": 이야기를 듣다가 보니까 어느 순간 잠에 들어서, "listening on and on, at some point I fell asleep". Treat 다 보니까 as a fixed phrase; it is everywhere in spoken Korean.