Why did the removal of wolves affect the entire Yellowstone ecosystem?

Removing wolves from the park affected much of Yellowstone because wolves are top predators and arguably keystone species.

Predators are often very important to an ecosystem because they control population numbers of other species, mainly their prey. Think of a very simple food web where birds eat insects which feed on plants. If there are no more birds, no insects will be consumed, leaving more insects alive in the food web. With more insects alive, they will eat more of the plants.

This same concept applies to wolves and Yellowstone, except the food web and effects of wolves are far more complex. Wolves feed on elk, and without the wolves, the elk population exploded. The elk fed on young aspen trees, so the park had very few young aspen trees.

Without the predation of wolves, the elk remained in one place and fed on vegetation by the rivers, which had tremendous effects. With significantly less vegetation, the riverbanks began to erode and the rivers widened. The temperature of the river warmed because there was no shade cooling the river, so the abundance and distribution of fish species changed. Birds that nested by the river no longer had a riverbank to build their nests on. Beavers used willow trees on the banks of the river for their dams, but there were no more willow trees by the river because of the elk, so the beavers disappeared.

Before removal(simplified):

Once wolves had been removed(simplified):

To read more about the effects of removing the wolves, see this link from Yellowstone National Park on reintroduction, this article on the controversy surrounding the reintroduction of wolves and if this has saved Yellowstone, or this link on the beaver-willow lack of recovery.