Draft means to draw, both in the sense of sketching an image onto paper, but also in terms of pulling — a draft horse draws a wagon, a draft of air is drawn into your lungs when you take a breath.
A cold draft is a current of cold air being drawn indoors. To draft someone into the army means to draw them into service, or make them serve. When you are drafting, or writing, a draft of a paper on the military draft in the U.S., you want to let your ideas flow freely. Then, in your next draft, you can work on filling in holes and edit the language down to what you really want to say.