붙이다 is 'stick/attach', and after a -아/어 form it says that the first verb's action ends with things joined together. Two combinations dominate these lines: 이어 붙이다, splicing pieces end to end (audio takes, or memories), and 밀어붙이다, pushing something through by force until it happens. Others are literal — 잘라 붙이다 (cut and stick), 연달아 붙여서 (run performances back to back).