Pin elastic to wrong side of waist casing allowance, along its outer edge. Align mid -point marks on elastic with center-front and center-back seams, and machine-baste elastic to waist casing allowance by sitching-in-the-ditch along pants seamlines from right side. Fold waist casing allowance and elastic to wrong side of garment and stitch casing in place through all layers along bottom edge of elastic, stretching elastic as you sew.
The "outer edge" to which they refer is indeed the cut edge of the fabric. I don't know which pattern you're following, but assuming there are only front and back seams, then do this:
*You have a circle of elastic, mark it in half.
*Pin it at one mark to the center-front seam (on the wrong side of the fabric).
*Then pin the mark across to the center-back seam.
*Use a long stitch to baste through the side seams (up/down, not across) so the elastic is attached at those points.
*Then fold it over so the elastic is sandwiched in the fabric and stitch through all three layers (fabric, elastic, fabric) all the way around. Stretch the elastic as you go.
If there are also side seams, then mark the elastic in quarters and pin to front, back, and each side seam. Baste through each seam, then sew through everything.
Clear as mud?
I've found it easiest to just separate the sentences and figure them out one at a time before going to the next sentence. Also, you're not required to do it the way they say. If you have a preferred way, then do it that way. They haven't yet sent the sewing police to my house and I almost never do it exactly the way they suggest!