The 다가 ending interrupts one state and switches to another, and here you were asleep or seated and then you get up. That is the whole of these lines: waking mid-sleep, sitting a while then standing, sleeping on the way and waking up already arrived, sinking into water and coming back up. 일어나다 also means to happen, but with 다가 and a sleeping or sitting verb in front it is the physical getting up.