Ive spent a lot of time at Lago deAverno (Lake of Averno) shooting dragonflies. There are several species of them but the most common is the Violet Darter in the photo. After a while you start to pick up on their habits and quirks, and you learn when you can get close and when you're wasting your time. The trick is to find one that's busy. If they are feeding, mating, or otherwise occupied then they are less likely to fly away. If they do fly off then just freeze -if the dragon comes back to the same spot (or close to it) then try again. If the critter lands several meters away from you then look for a new subject to shoot.
If you try to get close to a dragonfly and it takes off then simply freeze, or back up a step and freeze. If the critter comes back to the same perch then odds are it didnt take off because of you; it found something to eat and went after it. The trick is to move slowly and dont vary your pace. Predators often move slowly at first, but speed up for the strike. Photographers often move slow at first but get excited as they get close to the subject and the critter will take off because youre moving like a predator
Another technique that I've used in the past is the "stop and go" method: Take a few steps toward the critter, pause for a few seconds, take a few more steps, pause and maybe even take a photo (just to get it use to the flash), take a few more steps, etc. Sometimes you can get them use to you that way but it takes a lot of patience. Most of the mating shots that I have were taken by using the stop and go method.
Now that youre getting close the dragonfly is getting use to you, and its not viewing you as a predator. So take your time and compose your shots carefully so you wont feel the need to crop them later during post processing. If the dragon takes off just wait and see if it comes back. If you are very close (within inches / a few centimeters) then you might have to back up a little to get the dragonfly to come back to the same perch. Just keep your movements slow and even and when it lands move back in. Patience is the key to taking photos of dragonflies above life size get into too big of a hurry and youll go home empty handed
Dragonflies are such creatures of habit that when they take off you can hold your hand over their perch and sometimes they will land on it
I normally go shooting in the heat of the day, when dragonflies are the most active and when I have a better chance of finding one thats feeding (or a mating pare). If there is a particular species that wont let you get close, or that seems to be constantly in motion, then go looking for it early in the morning when its still trying to dry out from the previous nights dew.
I take almost all of my photos using either a flash as fill or as the primary light source, so the light from the flash doesnt seem to bother them. The only trouble Ive had with flash photography is getting so close to a dragonfly that I accidentally touched it with the flash ;)
I never wear bright clothes when I go shooting insects. I think it helps to blend in a little with the area where Im shooting, so I try to stick to neutral colors. Grey is almost always a safe bet, but any color that comes close to the reeds, trees, and other plants that they perch on will help.
Be patient, be persistent, and push your luck. For the shot included with this how to I was shooting with a Canon MPE-65mm macro lens set to 4x, so the distance between the front of the lens to the dragonfly was only 1.7 (4.3cm) minus the half inch that the MT-24EX extends past the glass. But I didnt start out shooting at 4x, I worked my way in from life size and I took photos as I got closer, and as the critter got use to me.
Practice, lots of practice... ;)