Translation:Could you clean up the room?
打扫 is the verb to clean/to clean up.
干净 is the adjective clean (describes a thing that is clean.) 打扫干净 is a verb + complement construction. Specifically, 干净 is a complement of result. It shows the result of the verb. 打扫干净 means to do the action of cleaning until the result is that the thing is clean. We don't usually speak this way in English, but it is normal to speak this way in Chinese. It might make more sense to an English speaker if you imagine a very messy room. 打扫干净 means "Clean until the entire place is clean."
Let's say you spend 15 minutes cleaning a room. The room is still messy but you stop there and watch a movie. You could say that you cleaned 打扫, but not that you cleaned up until the whole place was clean 打扫干净.
"This" is an extra presumption that isn't necessarily supported, I think. For example, the Chinese could be used in a situation where two people were talking on the phone about a room that needed to be cleaned before a guest arrived, so not necessarily "this" room, but "the" room that the guest was going to use.