Eagles demonstrate their role as ecosystem engineers through their nest-building activities by creating large and durable nests that serve as habitats for various other species after the eagles move on. Their nests provide shelter for animals like owls, falcons, and other birds. Additionally, their nest-building activities help in shaping the environment by influencing tree growth and nutrient cycling in their habitats. This demonstrates how eagles play a crucial role in the ecosystem by creating structures that benefit other species and impact their surrounding environment.