Essentially, the population that lived on Easter Island was trapped and eventually died off.
The people on Easter Island started creating the famous statues that I'm sure everyone knows about today, to help protect them from possible invaders. Once the people started settling in, they were provided with a number of natural resources and supplies, which unfortunately they had taken these resources for granted. Since they had been using all of the island's resources and using the many trees there for their civilization, they had no means of leaving the island once the last trees were gone. They were pretty much trapped on a dead island due to them causing the island to go through deforestation.