What's the deal with this eagle?

This is the view from the top of the tower just east of Bleak Falls Barrow. There's an eagle's nest, and an eagle that visits it. Climbing up and touching the nest has no effect, but shooting the eagle causes the nest to disappear.

Is the eagle there for a reason? Does shooting it have any effect?

Eagles are more than scenery in Skyrim. If you shoot them down with spells or a bow you can harvest their feathers and beak for alchemical purposes. But that is all.

You can also ride on their backs if you climb up to their nest and jump on when they land. It's cool to be flown around, but also very hard to stay on for long though! A well-timed Become Ethereal shout is the only thing likely to prevent gravitational death if you decide to take up hawk surfing.