I both agree and disagree with this. I don't believe there's any cosmic responsibility imparted towards humans that they must be caretakers of the Earth. To suggest so, to me, is the same sort of hubris which suggests the earth is made for us. I don't believe nature in any sense cares what we as humans decide to do.
I think it's in our best interest to minimize damage to our ecosystem, and I'm all for curbing needless suffering of both human and non-human animals. But nature doesn't innately care if an overspecialized and already evolutionarily limited animal like pandas go extinct. And we only care because it's part of a status quo we know and enjoy. And cute fuzzy animals tend to be more highly valued than, say, the less appealing key indicator species like various insects, amphibians, fish and reptiles.
If a human can assert that Because We Are, We Have Rights That Are Not Available to Others, that, too is hubris. Rights and responsibilities are human constructs (and perhaps of others as well...). Few of our Philosophers have ever focused on this aspect, because it tends to limit what we might be able to do by exercising our rights. I want the right to free speech...without the responsibility of being accountable for the consequences of my speech, for example.
In nature, every new life has the right to try to succeed; most never get to adulthood, and never get to reproduce (which is biological success). Each has the responsibility of dying and contributing to someone else's success--a responsibility that is enforced by others. If they do succeed, to become an adult and reproduce, they have a responsibility to give their offspring a chance to succeed...they cannot and do not horde all the resources, so they can be fat and happy and have plenty of reserves set aside against future need. Their responsibility is to take no more than they need, leaving the rest for others--and in fact, attempting to take more than one can use tends to make one less successful.
For example, the pair of Cooper's Hawks that have claimed the territory that includes our house have the right to select, and defend from others, if they can, access to the resources here. If they try to take too many resources, they will undermine their own (and their offsprings') ability to survive. If they try to control a larger area, they will find they are weakened because it takes much more energy to patrol a larger territory...
Humans have gotten rid of virtually all the limits in their pursuit of their right to survive and have offspring...and only a handful of humans today get to claim vast territories and resources, enforced by the social system, without responsibility to leave excess for others. The result is that some humans have excess, while most eck by with the absolute minimum that the few let drop, or that they can scrape up on their own.
