This attaches a verb to 하다 with -어서 so the first verb tells you how or in what state the doing happens: 나눠서 해 is "do it split up (between people)", 숨어서 해야 돼 "have to do it hidden", 신경 써서 했어 "did it with care", 이어서 해 볼게요 "I'll carry on from there". A few uses are just sequence rather than manner, as in 일어나서 하는 루틴, the routine you do after getting up. The -어서 is glued onto the verb stem and changes its shape, so it is rarely visible as a separate word. Compare -고 하다, which merely lists two actions in order; -어서 하다 ties them together, the first one shaping or enabling the second.
Register: everyday conversational; all the examples are casual speech