돌려보네 is 돌리다 (turn, rotate) + -아/어 + 보다 (see, watch) with the -네 (mild exclamation) ending. It probably means watching many movies like going through a rotation. So, 오늘도 난 영화를 돌려보네 = Once again today, I am watching movies in rotation. 돌려보다 can also mean "turning the roll of film" as they did in a movie projection room in the old days, but it is not likely what they mean.