치다 covers striking things of all kinds, and these lines spread across its senses: flicking (그걸 이렇게 돌려서 치는 병뚜껑 게임 'a bottle-cap game where you spin it and flick it'), clapping (박수는 즐거워서 치는 거잖아 'you clap because you're enjoying yourself'), and playing a keyboard instrument (피아노도 줄여서 치고). -아/어서 gives the manner or the reason. 손이 알아서 쳐 uses the fixed 알아서 'of its own accord'. Watch the last line: 치실 there is the noun 'dental floss'.