This was one of the harder parts. While the vest is inside out (with the liner facing outside) {edit: inside out, like normal, clothing inside-out from how you normally wear it. It's right-side out now, in the sewing sense, after turning in step 21. (thanks to BeanGolem)}, you'll notice that the last rough edges have the front outer and the back satin facing each other. Sew these two together, but don't sew the liner in with them. Try to sew upwards past the seam, onto the liner, so you're sewing the two inside liners together and creating a tube. Repeat for the other side.
When you flip the vest right side-out again, you should have a nice seam on the side, and an open pocket in the middle of the liner, on the inside of that seam. Sew this hole up with a "slipstitch", by hand.