Jon, that is a new side of you that you just revealed. Your landscapes are great, but this... I mean... wow PM me and tell me how in the hell u did that
There is not a secret recipe TC. Many species of dragonflies and damselflies have a tendency to circle around and come back to the same perch. I found that if I just hung out they would come back and pose for me. I shot around 10 AM to 2 PM in bright sun. They of course like marshes and bogs, and fields. The gear makes a difference. One guy I watched on Google uses a 300m f/4 lens, and a 100mm MACRO f/2.8 lens, which happens to be my setup. The macro is for closeup (3 ft and closer), smaller critters like the damselfly. The 300mm is for bigger dragonflies, and farther away, like 6 ft. or more. 2nd hand lenses can be had online for a lot less than new.
My macro was about $350: Tokina ATXAFM100PRON 100mm f/2.8 Pro D Macro Autofocus Lens for Nikon AF-D, Black
I use a 300mm Nikon lens that is pretty expensive but features a lightweight fresnel lens which makes it easy to handhold: Nikon AF-S FX NIKKOR 300MM f/4E PF ED Vibration Reduction Lens with Auto Focus for Nikon DSLR Cameras
Focusing is hard and requires a combination of autofocus and manual focus, make easier by the fact that you can wait for the critter to circle around and come back to you, and you will already be focused on his perch. I used f/4 to f/6 aperture, to get that out of focus look. Using a smaller f-stop is probably safer to ensure depth of field (getting mostly everything in focus from the wings to the tail). One guy says he shoots at f/11 for depth of field, but you can go higher.
The post processing was time consuming, as I took a LOT of shots, most were out of focus or bad composed. A lot of what you get is just luck.
I am sure that 46er could give you a lot of tips, since he is a much more experienced wildlife photographer than I am.
ALSO: Shutter speed should be a least 1/500 second or more, probably more like 1/1000 of a second or more. According to Google guys, some people shoot dragonflies with a point and shoot, as long as their lens has a macro function, so you do not necessarily need expensive gear.MACRO means 1/1 ratio, don't ask me to explain because I hardly understand it myself.