Sorry about the text. I was intending to have more of a tutorial but I ran out of time.
The real world shows you what the solution to the room is. You have to solve the puzzles in the dream world and progress through the dream world. Switching between worlds acts as a reset button in case you make a mistake since it resets the room and player position. I wish I had time to make this more clear, but before upload time I had been up for over 24 hours working on the submission and I needed to get it at least playable.
Thanks playing and the good feedback!