There are a couple of differences with the Klickfix equivalent, but here's my guess.
The first step with the fabric strap (*) might be easier done off the bike...
Facing the front of the bracket (where the basket will sit), locate and remove the four long bolts (in four corners). These go through the bracket into the two "C" shaped clips that surround the bars.
Remove C clips from back of bracket, and insert them over the handlebars, possibly with some old inner tube as a spacer/aid to grip.
Insert bolts and tighten up a little.
Rotate the bracket so that it slightly points up, and pop one end of the fabric strap into the slot to one side of the stem. Use the short screw, washer and flat plate to grip (pierce?) the strap.
Take the strap over the stem, and insert into the other slot, tightening the strap before securing it with the screw/washer/plate.
The strap should now prevent the bracket from drooping - rotate the bracket down as far as it can go (with a little pressure). If it is now pointing down, rotate up a little and tighten the strap further. If it's pointing up a bit, probably live with it, as it's likely to stretch in use anyway.
Finally, tighten up the long bolts to grip the handlebar.
I'm afraid I don't recognise the L shaped bits of plastic from the Klickfix equivalents I have. Perhaps they're spacers to accommodate different bar diameters?
Klickfix use a wire cable instead of a strap, but their instructions (which may be clearer than mine!) are at:
http://www.klickfix.com/images/downloads/bedlenk.pdf
Robert