Creating a Double Dug Garden Bed
Double digging to create a garden bed essentially loosens the soil to a depth roughly 2 shovels deep. By loosening the soil and incorporating organic matter such as compost or manure plants can root much deeper and have access to more water in the soil as the loosened soil acts as a sponge to hold more water. This allows for the cultivation of more plants in a smaller space and typically produces 4 times the amount of produce than conventional tilling.
The Technique
The watermark is very annoying in this video, but it adequately shows each step. You dig a trench to 1 shovels depth and then use a pitchfork to loosen the soil beneath that. Then you move a row and as you remove the soil with the shovel you place it in the trench you just dug.
Double Digging Part 1
I don't feel that they did an adequate job explaining what they were doing but it's good to see each of the steps.
A Slightly Different Technique
They appear to be adding nutrients to the soil after the trench is dug and then completely turning the soil with a shovel to another shovel depth rather than just loosening with a fork. They don't appear to be digging very deep however, and if it were done to a good depth may be much more work.