だが is a conjunction meaning “but” or “however”. It can be replaced with でも which is usually used in casual and colloquial speech by both males and females. On the other hand, だが sounds more like written language and it’s often used by males.
断る means “to refuse”, which is the dictionary form of 断ります. By ending a sentence with the dictionary form without any sentence ending particles such as よ or ね, he’s expressing his strong decision and declaration.
This phrase is so popular that it has become the subject of many parodies. So I will introduce how to use だが断る in a proper situation.
In this scene, Rohan being held hostage is told that if he sacrifices Josuke, his life will be spared. So he once pretends that he is inclined to do that, but then he says “NO” to astonish the enemy who was certain that the answer would be “YES”.
So it is used towards someone who gives you a very attractive offer when you are desperate, believing that you will say “YES”. Also, using this phrase after pretending that you are going to accept it will be more effective. For example, when the most popular girl in class tells you that she can go out with you and she thinks you will say YES as you look desperate, you can say; 本当に付き合ってくれるの？…だが断る。(Can you really go out with me? … But I refuse). Or when a rich person offers to lend you some money because you have no money and you can’t even buy food, you can say; 本当にお金を貸してくれるの？…だが断る。(Can you really lend me money? … But I refuse).