I have a hard time grasping the meaning of certain Korean phrases the first time I hear them, especially when they are quickly.
How can I politely ask someone to repeat something they just said?
"네?" is a semi-formal and probably the simplest way. If you talk to friends, you can simply say "응?". If you talk to someone really close to you, you can also say "뭐?"
There are a lot of ways to do this. But some very general, polite examples would be...
My Korean teachers always made me say:
다시 한번 해 주세요?
(Please say that again one more time?)
You don't necessarily need to ask directly. For example:
(죄송합니다,) (잘) 못 들었어요. ((Sorry,) I didn't hear what you said.)
There is no explicit request, but they will understand that you want to listen once more.
The simpliest way is to say "I am sorry. (미안해요 or 죄송해요)" first and then say "한번만 더요. (one more time, please)"
The more polite way would be
(죄송합니다). 다시한번 말씀해 주시겠어요? (I am sorry) Can you say one more time, please?