You slip the stitch
from the left needle to the right with the yarn in front (or just hanging down loosely at the edge) as if you're going to purl. That's putting the right needle into the stitch from the right toward the left. Then the yarn goes between the needles to the back so you can knit the next stitch. As long as you don't slip at both ends, it really shouldn't pucker the edge. Even if you keep the yarn at the back when you slip, it won't pucker the edge. It'll just give a bit of a bump on the edge.
Beautiful scarf! I love the look and techniques in Orenburg.