After capturing the animal, the next step in preparing the hides was to slaughter it. This is the melacha of shocheit. Practically speaking, shocheit includes killing a creature, even without causing it to bleed (such as by drowning or suffocation), plus causing bleeding, even without death.
Most of us don’t go around indiscriminately killing animals, but we must be careful about insects. The Talmud in tractate Shabbos (107b) tells us that killing a tiny insect on Shabbos is the same melacha as killing a camel.
The bleeding aspect of shocheit means that we should be careful on Shabbos not to do things that will cause bleeding. For example, if one is reasonably confident that picking his teeth will cause his gums to bleed, he must not do so. One must also be careful not to cause bruising, which is simply bleeding without breaking the skin.
As with trapping, one is permitted to kill an animal that presents a threat to human life, such as a rabid dog.
This is just an introduction to the concepts of the melacha of shocheit; it is not a substitute for a full study of the halachos.