Two big links between poverty and environmental degradation are subsistence farming and exploiting natural resources, particularly in the developing world. Poverty leads many in the developing world to become subsistence farmers, or people who grow just enough food for their family to eat to survive. In order to farm, these people must clear available land for agricultural purposes. Since they must inhabit and work unowned land this often means that they must clear jungle or woodlands, sometimes those bordering protected preserves. This cleared land scars the Earth through both the loss of natural habitat and also through erosion and water runoff of the cleared land.
The other big link between the two is the exploiting of natural resources. One story I read recently in National Geographic was that there are billions of barrels of crude oil reserves located under the heart of protected equatorial rainforest in Ecuador. Companies of course want to extract this resource for money. The government has held off from any drilling there since it is in important, protected land. But Ecuador is a poor country and the government wants the world to pay them to leave the oil alone and keep the rainforest intact. So they are using the environment as leverage to try to get money.