끊다 is cutting — most often cutting off a phone call, a stream, or contact — and the -고 clause is the last thing done before the cut. Several lines are about phone calls ('아니' 이러고 끊을 정도로, 쌍욕을 하고 끊었어요), and one is a host signing off a segment (소개해 주시고 끊겠습니다). The passive 끊어지다 is used when contact simply lapsed rather than someone deliberately ending it.