To get this achievement you would need to have a "Trigger Neosaur" spawn in the game, and be killed. It will only spawn for the losing team, the achievement, however, will unlock for both teams, as both teams are experiencing the "Dinosaur upgrade". (The advantage and disadvantages of the Dinosaur Upgrade) Thanks to Smudge Skyrider for the clarification on it working for both teams!
The spawn of a Trigger Neosaur depends on player progression (Mid to late game) for Neosaurs (Mutated Dinosaurs) to spawn in general (You may get in matches with no Neosaurs, depending on RNG and/or the progression of other players)
Note - The Trigger Neosaur is a catch-up mechanic, but it does not spawn in many games. As people have mentioned in the comments, people with many played games including those with 100+, and have yet to see one.
NutriWhip Commented that the losing team needs to be 2-3 objectives behind the enemy for it to spawn. (May increase the spawn chance, plus able to get 2 in a game)
NEW - I had a game where the enemy team had a Trigger Neosaur spawn but was only 1 objective behind. So either there is some randomness to it or it's predetermined if it will spawn in that particular match. I've noticed that it spawns in matches with the more rare and dangerous dinosaurs such as the Stealth Neosaur, Pyro Neosaur, and Umbral Neosaur! If your Dinosaur Cull objectives have these rare Dinosaurs, there's a good chance that a Trigger Neosaur could spawn! (It is common is be matched up with people with low progression, which means no Neosaur spawns or no rare Neosaur spawns)
The Trigger Neosaur looks like the Gas Neosaur (Red exploding one) and the Sniper Neosaur (Purple) but it is gold in color. When it spawns in the game, Leviathan will announce it, and it will be on the map as a gold dot.
What a Trigger Neosaur does is increase the difficulty of the dinosaurs and hinders the opposing team in some way. When you kill it, your team has a brief time period to accumulate a score which in turn increases the difficulty of the dinosaurs. (The difficulty level goes up to 5)
Description from the official site -
“Trigger Neosaurs, which are special dinosaurs with a golden sheen, may appear during a mission.
Defeating a Trigger Neosaur strengthens the dinosaurs fighting the enemy team for a set period of time, making mission progression that much harder for them.”