There are several ways of answering the question why do we value species? The first is utilitarian, and economics is catching up in how we can measure the value of the benefits that nature provides to people.
Two years ago, the Nature Conservancy of Canada partnered with TD Bank to identify the natural capital values of forested areas across Canada. There's a lot of variation, but the average value of a hectare of forest was over $25,000 per year, and if it is maintained, it will provide those services forever. Even the TD economist recognized this is a gross underestimation, that there's only so much they could value as economists, and the true value was much higher.
From a utilitarian perspective I think the main reason we don't want to lose species is that we simply don't know what the impacts will be of losing those species directly, or how those impacts can cascade through the ecosystem and affect people.
I think the other reason is the intrinsic values, as Dr. Kerr alluded to. Do we want live in a world where we're consciously choosing to let species go extinct? I cut the past generations a little slack because they didn't have the science, the knowledge maybe, that it was the last place or the last example of a species. We don't have that slack now; we know which species are declining and are unlikely to be passed on to future generations. It's really our decision what we're going to save. In some cases, we need to make those decisions very quickly.