Why should we care?

We shouldn't care about plants and animals going extinct because it really has no effect on us. Huh? Where did you go to school? What was your major? I see. Did you have to take any biology courses? Oh, well that explains everything. It seems self-evident to me that our environment includes us, and up until I started wrangling with politicians I thought it was fairly obvious to everyone. It seems it's only obvious to those of us who aren't engaged in oil drilling, strip mining, or money hoarding. I guess the whole acid rain thing a few years ago didn't wise anybody up. The polar bear is only one of a few species being affected by environmental degradation; it just happens to be the most visible.
Salamander numbers have been declining. The way the salamander is built, it needs to keep its skin wet and cool, which it does behaviorally. Unfortunately, its skin is more permeable than ours, so it ends up letting in a lot more impurities than the rest of the body is prepared to deal with, and the animal dies. This phenomenon has also been noted in frogs. (Quick, what's the difference between a frog and a toad? Toads can live in arid places because they have thicker, less permeable skin.)
At the rate the rain forests are disappearing, we will not get to test most of the plants for pharmacological compounds. There is a certain periwinkle in the Amazon that has shown promise as a treatment, if not a cure, for cancer. We can't replicate that compound without intensive study of that flower, but if the rain forest is cut down the flower dies, too. Besides, the forests in themselves are pretty neat. They generate a lot of their own rain through evapotranspiration. And trees--yes, I seem to recall something about trees using carbon dioxide. (They'll lose that ability when they die, obviously, but there's a lot of living stuff with a need for carbon down there.)
Fish in the ocean are dying because the ocean water is turning acid. The ocean is a carbon sink, but it is fast reaching its saturation point. I doubt all the radioactive heavy metals we put down there in cement bins is helping, either, especially not now that the bins are starting to leak. We've had a lot of strange-looking critters turn up because of living in a polluted area. Given what radioactive poisoning has done (look at Marie Curie), I'm not surprised that the people of Nevada aren't particularly enthused over the idea of storing radioactive waste in Yucca Mountain.
And, of course, there are the polar bears. When ice starts melting at such a rapid rate, it should be a clue that something isn't right. Dying salamanders, drowning bears, deformed babies, declining pikas--these might all be taken as an indication that something is wrong. So why should we care? We should care because we live in this world, this environment, too, so whatever is not good for other living things ultimately is not good for us.