Oh the paper is to make them be hexagons. It's a technique called English Paper Piecing, and it's a way to make sure the corners stay corners. It's apparently quite an old technique-- they found an example of it in progress among Emily Dickinson's papers, she'd cut some paper up into scraps and had a set of six of them made of scraps of fabric, including some of the same fabric a figure is wearing in a photograph that they figured out had to be her because of this.
If you just piece hexagonal bits of fabric together they tend not to keep their shape, according to basically all the tutorials I found when I searched for more information about it. So... the accepted practice is just to bind your fabric around paper templates, and then join them together, and then when you've got them all together you can go back and pop the paper back out.