There are no natural systems untouched by human actions, and there is no way to separate the consequences of a changing environment like extinction from human activity.
Conservation is a movement to preserve the natural systems that were here before industrialisation, and are here now, not to "let nature run it's course*.
what right do humans have to intervene in this?
There are no rights in nature, rights are a part of the fabric of society. We want to conserve nature and so we will.
Human society exists entirely within nature. There's not some diving line where this place over here is natural and this place over there is unnatural. We decide what rights we have to what, because we decided that rights should exist.
There is no leaving well enough alone, unless you are content to watch it decline into deserts and polluted wastelands. Every action has consequences, and since we have already acted, the state of nature is the consequence of our action whether or not we continue to intervene.