It's always reassuring when scientists use descriptions that actually sound like something I've experienced. So when researchers at the University of Wisconsin describe the lack of sleep as like "feeling that your brain is full of wool", they have my full attention. I have been that wool head.
Sleep creates the space needed for learning new information, they describe it as "spring cleaning" the brain so that there is a nice tidy place to put new things. Without sufficient sleep, the information received by the brain today is competing with the noise and clutter from yesterday. The brain, without sleep, is in no fit state to learn anything new.
The sleepless brain is the desk where you can't find anything. Sleep de-clutters the brain. Sleep makes the path for learning.