The carbon dioxide molecules exist in the atmosphere, and they are responsible for pretty much all life on the planet, including plants, animals and even humans.
Carbon dioxide molecule
The carbon dioxide is then absorbed by the plants on Earth, that use them to make carbohydrates for themselves. They use the carbon in a proccess called photosynthesis, where oxygen is produced, which is what we breathe.
Herbivores in the same ecosystem as the plants eat the leaves and sometimes even bark of the tree, so some of the carbon gets released into the air, but most of it is absorbed by the animal.
When the herbivore is hunted down by a predator, or in this case predators, some of the carbon dioxide gets passed on to the predators, and some goes up into the atmosphere again.
When the dead animal starts decomposing, the carbon that is left in it's body either goes up to the air or into the ground.