뜨다 is rising or floating — something getting off the ground (어느 정도 뜨다가 안 뜨고 뜨다가 안 뜨고, rising a bit then stalling) — and in 눈을 뜨다 it is opening the eyes: 잠깐 눈 감고 뜨면. Several lines in this sample are lookalike verbs rather than 뜨다: 떠들다 is chattering away (쉬지 않고 떠드는), and 떠먹다 is spooning food up.