(I think your picture is upside-down)
I have one of these (Brake! LED taillights) and I really like it. But it was a pain to install, as you're discovering.
Have you removed the bulb sockets from the taillight assembly? A half-twist each and they're out, with the bulbs still in them. Remove the bulbs; they won't fit anymore with the LED board in place. One of the sockets will power the new Brake! light, the other will remain empty, but you still need to re-install both of them to keep water out of the taillight assembly. (when you get to the re-assembly point)
See the two screws on each side of the taillight? They're near the rack holes in your picture, way up at the top (bottom of your picture). Try removing those two screws (don't lose the washers). They each go thru a rubber grommet that cushions the taillight assembly. The whole unit should be able to be pulled out at this point.
Then it's time to heat up the oven, but that's another adventure...
Make sure you pick a flashing pattern you like before you button everything back up. Happy flashing!