-(으)면서 means "while, at the same time," and 느끼다 is both physical feeling and coming to realise something. The stock phrase is …면서 느낀 게, used to introduce a reflection: 오늘 첫 메뉴를 먹으면서 느낀 게, 선배한테 죽어라 구박받으면서 느낀 게 뭔 줄 알아요?. 듣다 is the most frequent partner here — 들으면서 느끼는, 지금까지 들으면서 느꼈을 것 같은데요 — and these lines are mostly people talking through an experience on camera. The realisation sense, not the sensation sense, is what carries most of them.