There are three primary ways oil injures wildlife:
The oil gets on the fur and feathers and destroys the insulation value. Birds and mammals then die of hypothermia (they get too cold).
They eat the oil, either while trying to clean the oil off their fur and feathers or while scavenging on dead animals. The oil is a poison that causes death.
The oil impacts them in ways that does not lead to a quick death, such as damaging the liver or causing blindness. An impaired animal cannot compete for food and avoid predators. Oil also affects animals in non-lethal ways such as impairing reproduction.