That's adorable! I'd use the two sided fusible webbing, because I'm lazy. Then you can peel off the paper after cutting out the shapes and iron them onto the fabric.
If you do that, you don't technically have to stitch down the edges unless you want to. I have two small wall quilts that I just ironed on shapes, and they've both held up (no washing) for 5+ years and counting. Stitching the edges is just extra insurance if you want it to last 20 years, or expect it to get handled.
I would bet that she used the fusible-it's so much easier than trying to hold the shapes in place while sewing them down.
There's a tutorial here: http://sewmamasew.com/blog2/2008/07/summer-sewing-simple-machine-applique/