How do I get the feathers?

In order to obtain the feather, you first need to purchase the Mighty Eagle upgrade. That unlocks the Mighty Eagle, which you can use to perform massive destruction in any level.

Using the Mighty Eagle automatically completes the level, and the feather is filled up based upon the number of points you scored. If you do enough damage and get enough points, the feather is filled to 100% (Total Destruction) and the feather is shown on the stage select for that level.

The Mighty Eagle can always be used on levels you have completed, and it can also be used to bypass levels that you find difficult, though this use is on a timer - you can only use the Mighty Eagle to skip a level once an hour.