해내다 is not merely doing something but getting it done despite the difficulty — pulling it off. The -게 slot says how it looked: 손쉽게 (as though it were easy), 완벽하게 (perfectly), 능숙하게 (expertly). 멋지게 해내다, pulling it off brilliantly, appears three times, twice in captions wondering whether the performers will manage it.