Writing down your thoughts is a kind of hill climbing process. As you dump your thoughts out onto paper you free up your mind to think about a different part of

Writing down your thoughts is a kind of hill climbing process. As you dump your thoughts out onto paper you free up your mind to think about a different part of the problem / the bigger picture / the finer details. But your entire thought process up to that point (including the dead ends you crossed out) is still there in front of you as an external structure of concepts that can be built on. The more open and "invisible" the tools are the better, and there are few that can match a simple pencil and paper.