Background from the Silmarillion helps explain this.
The reason why the Eagles refused to become involved is because they are representatives of Manwe, king of the Valar (think of the Valar as something between gods and angels). Twice before the Valar had directly intervened to save the mortal races from the original Dark Lord Morgoth, both times with unintended consequences. This time they wanted the mortal races to rise to the challenge and defeat the Dark Lord themselves.
The reason why the Eagles refused to become involved is because they are representatives of Manwe, king of the Valar (think of the Valar as something between gods and angels). Twice before the Valar had directly intervened to save the mortal races from the original Dark Lord Morgoth, both times with unintended consequences. This time they wanted the mortal races to rise to the challenge and defeat the Dark Lord themselves.