You'll have to be careful you don't get hurt doing this; remove one strut and the whole thing comes crashing down, so have help and a long 2X4 to help prop the thing up.
On each end, there is a black socket which engages onto a steel ball. You'll notice on this socket that there is a curved metal strip, the metal being about 1/4" wide, maybe a little narrower than that. On the socket also by the strip, you'll see a small "notch". You can insert a small straight-blade screwdriver into the notch and put the tip under the metal strip and just twist/pry the metal strip "up" away from the rest of the ball. You don't need to remove this strip, just pry it up and at the same time pull the socket off of the ball. (You'll need about 1/4" gap under the strip)That end of the gas strut will now be disengaged from either the body or tailgate, depending on which end you disconnected first. You'll notice the new strut came with the bracket, the bracket I believe bolts to the gate side. If you want to use the new bracket that's fine, then don't unclip that side, unclip the other side and unscrew the bracket. I prefer to just unclip both sides (it's easy once you've done it a few times) and remove the new bracket form the new strut, then just clip the new strut onto the old (still attached) bracket. The new one you can just push onto the balls, no need to mess with the clips on the new one.
But like I said, as soon as you disconnect just one strut, the gate WILL come down, so be prepared, have a friend AND something to help prop it up. To do the first unclipping, it may make the old one easier to disconnect if your friend can push up slightly on the tailgate to help relieve some of the tension on the socket on the end of the strut.

Gilly