The Economist makes the case:
[A]s people get richer, their interests begin to extend beyond necessities towards luxuries: for some people that means expensive shoes, for others a day’s bird-watching. Green pressure groups start leaning on government, and governments pass laws to constrain companies from damaging the environment. In the West, a posse of pressure groups such as Greenpeace and the Environmental Defence Fund started up in the 1960s and helped bring about legislation in the 1970s and 1980s.
Growth also has indirect benefits for biodiversity.
People clean up their environment in ways that help other species: through building sewage-treatment plants, for instance, and banning factories from pouring effluent into rivers. Prosperity and peace tend to go together, and conflict hurts other creatures as well as man, as the wars in the Congo have shown. Richer countries generally have better governments, and conservation cannot work without an effective state. Agricultural yields rise, allowing more food to be produced on less land. Population growth rates fall: in East Asia, fertility has dropped from 5.3 children per woman in the 1960s to 1.6 now.
Another Economist article notes that this process isn’t quick:
In its early stages economic growth often causes people to multiply faster as death rates come down but birth rates stay high, as is happening in Africa now. That intensifies competition for resources between humans and other species. But when countries become richer, more women get educated and take jobs, more people move away from farms and into cities and birth rates start falling.