Questions & Answers about je bangeun hangsang kkalkkeumhaeyo.
제 means my in a polite/humble way.
It comes from 저의, which also means my, but 제 is the much more common shortened form in everyday speech.
So:
- 제 방 = my room
Compare:
- 제 방 = my room (polite)
- 내 방 = my room (casual)
If you are speaking politely, 제 is the natural choice.
은 is the topic particle.
So 방은 means something like:
- as for my room
- my room, ...
In this sentence, 제 방은 항상 깔끔해요 sets my room as the topic and then says something about it.
A very literal breakdown would be:
- 제 방은 = as for my room
- 항상 = always
- 깔끔해요 = is neat/tidy
English usually does not mark the topic this way, but Korean does.