The Eagle's Nest is the location from which Stephen Colbert broadcasts his news show, The Colbert Report. Some 'locationistas' will say that it is filmed in The Daily Show's old studios, but from the window in the "Axe Body Spray Interview Zone", it is clearly visible that The Eagle's Nest is located on the top of the Statue of Liberty.
It is stylized to pay tribute to the 'heroes', those that watch the show, by honoring Colbert himself. After all, what better way to thank the heroes, than thank the person the heroes find heroic, Stephen Colbert?
The Eagle's Nest was not the name of Hitler's mountaintop retreat. That is a misconception spread by the terrorists. Hitler's mountaintop retreat was actually named the Bear's Den. Stephen Colbert named his studio the Eagle's Nest specifically to dispel this rumor.