Have you watched Amy's "Demo of a small project" video? At the very end she shows and discusses the last stitch.
After you knit and slip all the stitches you are left with that last one on the right needle. Basically you cut your yarn, remove the needle from that last stitch (don't loose the loop though), then you'll secure the yarn by running the free end (that you just cut) through the loop and pulling snug. You'll have to work that tail in for few stitches more so it doesn't come loose, but it won't just unravel anymore.