Why do humans care so much about killing livestock humanely, when animals in the wild just kill other animals brutally by nature without caring?
Humans are empathic. This means we mirror others' emotions and sensations when we witness them. We're not the only animals with this capability, but what it means is that we understand when we're causing pain, and doing so deliberately constitutes cruelty.
Cruelty is a trait which, when practiced by humans, tends to lead them to be cruel to one another as well as other species. Therefore it is highly antisocial and not conducive to a peaceful and healthy human society. As with empathy, cruelty is not something present only in humans, but only a few other species have demonstrated the self-awareness and empathy necessary to engage in cruelty.
Human emotions are therefore the primary reason for this behavior. Because we feel better about animal slaughter when we know the animals did not suffer, we're more comfortable with buying products made from them. So this is an excellent marketing tactic, and a good way to preserve the industry. Remember: HUMANS ARE NOT ALWAYS RATIONAL. Repeat that several times for emphasis. The reason why humans CARE about things is rarely based on logic. The fact that it's not logic-based changes nothing.
Finally, stress in animals can alter the flavor of the meat. This alteration is so extreme in some species that it changes the grading of the meat. Stressed pig meat cannot be sold as premium, and is fit only for ground pork products and pet food. This is no small consideration.
Now, the above addresses slaughter only. Of course, treating livestock well prior to slaughter means healthier livestock. Healthy animals put on and keep on weight better, and are less likely to pass on zoonoses. They are even more nutritious. So it's utterly illogical to mistreat livestock.