It was super easy to do. First, using my upholstery gun, I stapled a layer of thick batting onto the bulletin board.
I cut my burlap, making sure that there was enough room to wrap it over the batting and securely staple it behind the board. I also glue gunned the corners of the burlap on the back of the board because the staples weren't long enough to go through both the burlap and the batting in the corners.
Here's the finished product!