The hunters in this story are using a time machine to conduct a "time safari" - going into the distant past to hunt animals that are extinct in the contemporary world. However, their mere presence in the past may be enough to alter the flow of time - after all, the "normal" flow of time did not involve hunters time traveling and killing things according to their whim.
The hunters are only allowed to kill certain creatures because it isn't clear how their actions will affect the future, if at all; it's possible that by killing the "wrong" animal, they will alter the course of evolution and create changes in the timeline that they cannot predict based on a single death. Therefore, the safari guides scout the area beforehand, finding animals that would have died of natural causes and marking them for death only minutes before they would have died anyway. Presumably this, along with other preventative measures like the antigravity Path, will minimize the impact that the hunters have on the timeline.