This is the 4408th most frequent Korean word.
한다며
"한다며" means "saying/claiming that (someone) does/will do (something)."
In this sentence, '한다며' is used to report someone's speech or assertion made prior to their action of leaving.
그는 오늘 일찍 잠자리에 든다며 나갔어요.
He said he would go to bed early today and left.
Here, '한다며' questions a prior statement or claim made by the other person, expressing doubt or requesting clarification.
네가 운동한다며? 매일 아침 운동하는 걸 본 적이 없어.
You claim that you exercise? I've never seen you work out every morning.
In this example, '한다며' serves to indicate a reported speech while summarizing the promise made by the friend.