Maybe I'm a little late. I'm no pro, but having been wiring some lights up to my Vee using the version 1 adapter today, I'll try and help.
I'm getting some info from here:
I'm also going to make the assumption that the V-strom is of the Non-CANbus variety, so the diagram is at the bottom of the PDF you linked. Those instructions appear to be pretty confusing, and somewhat vague, so I'm going to do my best using the that diagram to what I know the EB adapter wires to be.
The Vstrom's ground wires all go to the negative battery terminal, so the black wire of the Coocase plate should go to the black wire of the Eastern Beaver adapter.
It appears the white (tail) and red (brake) wires of the adapter should both connect to the yellow wire of the plate.
So I suppose you should connect the red wire to the positive battery terminal, according to those instructions... hopefully the wire is long enough, but it should be easy to route it next to the under-seat tray.
If you've got the V2 adapter, you should connect the purple to the red and white as well.
The blue and yellow wires in the adapter are for turn signals, just wrap the ends in tape so they don't short to anything.
Hopefully that was helpful, but that's the best that I think that I can do.