내뱉다 is pushing something out of your mouth, and in these lines it is nearly always words being let out in one go. The -게 adverb sets the tone they come out in: 담담하게 내뱉는 'saying it flatly', 해맑게 내뱉는 걱정 'a worry blurted out brightly', 세상 냉정하게 내뱉은 한 마디 'a line delivered ice-cold'. It suggests the words came out without being weighed first.