"Inherit the earth" comes from the Bible, which says "The meek (people who are not proud and who are humble before God) shall inherit the earth." In the Bible, it means that one day the strong and proud will no longer be in power, and the people who are humble and fear God will be in control of Earth. In your sentence, the meaning is a related but different. If a nuclear (or other) doomsday comes that destroys people, spiders will still survive and will thrive.