Why do they exist and have a seeming mission to survive - because life's a struggle. A struggle to find & keep territory, find food, develop to maturity, find a mate & reproduce. Nature selects those individuals that can best do those things ('survival of the fittest') and their genes are passed on.
In the West for most of us, life isn't a struggle. Sure, we have tough times - trying to find a job, pay the bills, escape from the WT, etc. but we don't starve or die of exposure. We have cheap food, heating, running water, shelter, etc.
It can be difficult to appreciate just how tough wild animals have it. If, say, a leopard breaks its leg, then chances are it will starve as it cannot hunt or drag carrion up a tree to safety. There are no vets in the wild.