If you do not have the stock Vespa folding rack or grab rail, you cannot install this backrest.
Along with the backrest itself, you should have gotten:
two rubber-covered metal clamps
two 6mm 50mm-long screws
two 6mm 34mm-long screws (or similar--I honestly don't have the originals)
two 6mm hex nuts
You'll also need:
4mm hex key
your preferred wrench or socket
Optional but extremely helpful:
two Velcro or zip ties
two 6mm 25mm-long screws (instead of the 1.5-inch-long ones)*
two 6mm acorn nuts (instead of the hex nuts)*
a few choice curse words
The install will go much easier if you open the seat, and keep the rear rack in the open position. I reversed the springs and keep my rack held down with a hose clamp, but a simple Velcro or zip tie will work temporarily.
Remove the two screws holding the rear rack to the grab rail. Be careful not to push down on or set anything on the rack at this point; it is now only held on by the 2 screws into the frame.
Set up the backrest with the upper bar in front of the top of the rack support, and the lower flat bar behind it, resting on top of the rack or its springs. A Velcro or zip tie (not completely tightened) can help hold the backrest in place. A new backrest should come with a little rubber button that sits between the backrest bar and the upper rack support. If it is missing (mine disentigrated after just a few months), I suggest putting a couple of small adhesive felt furniture pads in between the chrome pieces to prevent them from scratching each other. Alcohol will easily take the adhesive off if you need to remove them. You shouldn't need anything between the lower part of the backrest and the rack.
Put the two clamps on over the top bar of the backrest and the rack support. I found a pair of pliers helps open the clamps up enough to get them over both pieces, and close them down again once in place. If for some reason you don't have the clamps that should have come with the backrest, you could substitute a couple of small hose clamps. Just be sure to cover the hose clamps or the bars with something to prevent scratching the chrome. Do not tighten anything up yet.
Install the longer screws (plus washers) through the lower backrest bar, rack, and grab rail. This will be the most difficult part of the installation. It is generally difficult to get everything to line up perfectly on the first shot, so if you do it, celebrate! If you're like the rest of us, just take your time, yell at it all you need to, and be careful not to strip out the inside of the grab rail. Once you're satisfied you have both screws lined up and they've caught inside the grab rail, you can go ahead and tighten both of them. Because the backrest is not angled at the screw holes the way the rack is, you will likely be able to see some screw threads between the backrest and the rack. That's fine.
Push the two clamps as far towards each side as you can while still keeping them fully upright on the backrest bar and rack support. Tighten them up with the shorter screws and hex nuts (or just tighten the hose clamps, if you went that route).* Take off any Velcro/zip/other ties you used to hold the backrest and rack in place.
You're all set! Close the seat, grab a passenger, and ride off into the sunset! The backrest is also yet another useful point to bungee bulky items to. Your rear rack will remain available to use for whatever you wish.
*I personally recommend using slightly shorter screws than the ones supplied, as well as acorn nuts instead. Depending on which direction you install the supplied ones, it is possible to scratch your seat and passenger, or whatever you put on the rear rack. Using shorter screws and acorn nuts eliminate this problem. Note the very small margin of tolerance between the backrest and the seat.
Bonus points: The backrest only comes in black vinyl. If you have a tan, Super, leather, or any kind of seat other than black vinyl, get a custom cover for the backrest so it matches.