붙이다 is sticking one thing onto another, and 갖다 붙이다 is 'bring it over and stick it on': 꼬리를 거기다 갖다 붙이면 'if you stick the tail on there'. Figuratively it means forcing a connection that isn't really there — 너무 갖다 붙였나요? 'was that too much of a stretch?', 참 여기저기 갖다붙이기 나름이다 'you can attach it to anything you like', 왜 다 병을 갖다 붙이냐 'why do they slap a disease label on everything?'.