The Eagle is a symbol of Divine Justice. Appropriately, it is formed by the souls of just rulers in the sphere of Jupiter. How do we know this? Well, a major clue is that the Eagle talks about Divine Justice, emphasizing how incomprehensible God's justice is to man. Also, at the end of Canto XIX, the Eagle blasts a huge list of corrupt or unjust kings.
The Eagle, as a symbol, has a long history. But the one most relevant for our discussion is the golden eagle of Rome, carried into battle as the Roman Empire's emblem. So how did "Rome" turn into "justice"? Well, remember Justinian? His name should give you a clue. His major contribution to history was codifying and improving all of Roman law. While Rome was known for its conquering heroes, a lot of its conquered peoples found their lives improving after being put under Roman rule. And the 'golden' part of the Eagle is kind of carried over to our Eagle in Canto XVIII when we realize that it is formed by these glowing bundles of lights called souls.
Consider Canto XX, where the Eagle introduces the six souls that make up its eye. You say, why the eye? What does an eye have to do with justice? Well, vision is the one thing in Heaven that can win you more of God's love. Beatrice highlights that Dante needs to perfect his vision. Once one can see things clearly for what they are instead of for what they seem to be, it becomes easier to judge them and assign rewards and punishments to them, (i.e. what the law does). That's why the souls in the Eagle's eye are accompanied by the refrain, "now he has learned." Learning – to Dante – is a process of improving one's vision. Now that these souls have learned what's virtuous, they change their behavior to conform to that idea. This is what gets them into Heaven.