Because the rate at which offspring are produced in every species is greater than the rate at which the environment can provide food, shelter, and other needs, individuals who carry advantageous traits will come to outnumber those without them, causing a shift in the common characteristics of the species over time. This is part of the definition of natural selection.

All species have adaptations that help them to survive in their environments. These would have appeared due to natural selection, because they conferred a survival advantage. They might even include, for some species, ways of limiting reproduction rates when resources are scarce. The factors above tend to limit population growth; but unless a species is populating a new ecological niche and there's plenty of room in which to grow and spread, then there will always be population pressure. If this wasn't the case then natural selection would not occur, and we would not see evidence of species having evolved over millions of years to become more adapted to their environments.

I'm not sure what the confusion is here. Surely an example of human settlements overpopulating an area, outstripping the resources, and then facing disaster, as appears to have been the case on Easter Island with the statue-builders, would be enough to illustrate the point.