Hi Brandon,
The double refers to doubling back and threading through the eye twice in the way that some anglers use for a Trilene Knot:
That is the wrong knot admittedly, but it shows the process clear enough - just translate that to a Uni. There is no double wrapping - only a double feed of the leader through the eye. With stiffer leader, it can sometimes be a little tricky getting the double to stay in place on the swivel while forming the knot, but with the supple leaders typically used for GT fishing, it is OK with a bit of practice.
I should also admit that I was sceptical about the necessity of this until this year. When fishing heavy drag (about 15 kgs), I lost the fish of a lifetime to the very issue which Ed had warned about. 170lb Varivas Shock Leader sliced by the eye of a 3/0 NT swivel. The fish hooked up and as soon as it ran, ping and I was loose. The leader came back with the Uni wraps intact but the section which was threaded through the swivel was cut. Later the lure (a Seafrog!), floated back to the surface with the swivel and split ring still attached.
I think this is only a problem when fishing super heavy drag, but since then I have made a habit of double threading leader through the eye of the swivel.
I hope that helps explain a bit.