When you start hunting, you gain 1 point for the animal shot and one point for a successfully kill using the weapon. These are called Ranks.
All ranks are limited to a maximum of 1000 points, aimed to make the system as accessible as possible. Hence, once you kill more than 1000 of the same animal, or once you use the same weapon type for more than 1000 kills, no further hunter score will be added to that specific rank.
You can find more details in this post about the Hunter Score and more in depth info in our Wiki as well.