“There are multiple reasons that people give. One is that some people do not think that a world filled only with cattle and pig farms and other such animals, but no other kinds, is a good place to live. This may be thought of as viewing the world as art, rather than as a factory. What do you want to see out of your window, and what do you want to see when you go on vacation? We could, if we wanted, divert the Colorado river and use the Grand Canyon as a giant landfill for our garbage. Is that a good idea?
Another thought is that we sometimes develop medicines and other useful things from different species, and without those animals, those new developments will not take place. And since it is impossible to know whether or not some species will prove useful in the future, preserving them all, insofar as one can do that, is the best way to have the most options in the future.
There is also the argument that is easily seen from thinking about an analogy. In the past, miners used to take birds down with them into the mines, and if the bird died (which was more sensitive to contaminants in the air than people), the miners would leave the mine. Without the birds, they did not know there was a problem until a person dropped off, and then it would likely be too late for them. Thus, the birds were an early warning system for danger in the mine. Likewise, animals may be viewed as an early warning system for life on planet earth; as we pollute the planet, the more sensitive animals die off first. And, of course, there are health consequences before one dies, so there is reason to keep the planet cleaner than just clean enough for humans to live. If we preserve the planet in a condition that saves the more sensitive animals, it will be safer for us.
Obviously, some people find some arguments more compelling than others.” (source)
This is a set of fairly good reasons. I haven’t come across good reasons for it before. Mostly what one gets is a bunch of appeals to nature (fallacy).