The basis for arguing for action on climate change is the belief that we have a moral responsibility to people in the future. But this is asking one group of people to make wrenching changes to help a completely different set of people to whom they have no tangible connection. Indeed, this other set of people doesn’t exist. There is no way to know what those hypothetical future people will want.
The premise is that by most models, we'll all be long dead by the time climate change is truly catastrophic for humanity. Our descendants won't know or care who we are. If you could go back 300 years, what would you ask people to sacrifice for us?
I find this interesting. The article isn't saying we shouldn't care, just asking why do we care? As an aside, and perhaps unsurprisingly, our actions ("revealed preference", as economists call it) suggest that we don't care about it as much as we say we do, but to the extent that we do, one possible explanation is that the very abstract notion of 'survival of the species' is actually important to us.
I think one flaw in the question though, is that the premise is weak: we can see climate change affecting the world now, and we can see enough of it to worry that people closer to us (our older selves, children etc) will be affected by it. And even if the consequences were further away, the scary part would be finding out that we've reached a point of no return in our lifetimes. That'd be a lot of guilt to deal with.