How can I use a writer's notebook to improve my writing?
To add on to Psicofrenia's excellent answer: Simply put, a writer writes. All the time. If you're not at your desk, you're still writing in your head. The notebook gives you somewhere to put your thoughts for later perusal.
So if you're sitting at a subway station waiting for the train on the way to work, you may observe two really interesting people having a conversation. You write that down. A year from now you need some background characters to liven up a scene, and you consult your notebook for an example of real people doing real things — how they look, how they carry themselves, the cadence and slang of their speech, and so on.
If your train trip takes you an hour, you have a notebook to write down the great plot bunny which attacked you on the way up the stairs. Or to write a description of just how the morning light hits the bridge on a hazy day in June, and how far into the city you can see before the buildings become indistinguishable. This is both recording real detail for later and practice for thinking about landscapes and how the weather affects them (so that when you write a fantasy story and your heroine is standing on a mountaintop in early summer, you will know that in fact she can't see twenty miles south).
When you practice observing, you get better at it. Writing down your observations forces you to go into more detail. You then get better at inventing details, because you know what details exist in reality.
I get entire scenes, pages of stuff sometimes, when I'm on the treadmill. The notebook is the place to jot that down, so that when I get out of the shower I can incorporate it (or not). It saves me from having to memorize three pages of really good interaction through the shower and a phone call and cleaning up cat puke and two emails and hey, my auction finished and now what was that scene again?
Having these details, these thoughts, these exercises available to you later enriches your writing. You don't have to rely on making up every bloody detail or remembering through the mists of time or researching June sunlight at a particular latitude in certain weather conditions. The things you write down in your notebook give life to your writing and keep your mental faculties sharp.
Basically the idea is to take notes. Supposedly, a writer is always receiving insights of creativity and new ideas. If you have a notebook, you can write them down and not forget even if you are in the middle of street or something like that.
The other function of the notebook is to collect interesting things. If you know how to draw and cross a interesting fountain, you may want to draw it or write about so you can see if you can use it later in your works.
So, what a notebook can do for you is allow you to keep notes for a later use.