Follow this link to the Injury and Recovery section of this web site for a thorough answer. 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.
A professional team and dozens of volunteers, including veterinarians, set up a cleaning facility and recovery facility. Dawn dishwashing detergent was the cleaning agent of choice.