If you have a part of a roof, that part is your room. Just imagine a roof divided into sections in a house, with the sections going all the way down to the ground. Each section is its own room, right?
The reading of 部屋 is a bit unusual. Both 屋 and 部 uses kun'yomi readings. It's actually read as へ, due to something called ateji. Ateji is when kanji are used just for their sound, regardless of meaning; or just for their meaning, regardless of sound. In this case it's the former, as 部 was chosen for its archaic べ reading, which was close enough to へ. This is all a bit much, so here's a mnemonic to help you remember this reading:
Just imagine that you rent out a room and the only one they have left is the hay (へ) room in the barn, so you sleep there for the night. Imagine the smell of hay and how scratchy it is to sleep in this room.
There's a bed in my room.
The room is facing south, so it gets good sunlight.
I plan to clean the room this afternoon.