Why does the eagle represent the US?

Apart from the eagle holding a snake, there is also a cactus.

The settlement of Tenochtitlan (which was the capital of the Aztec Empire and currently Mexico City - Mexico's capital) occurred after an omen took place. This omen was an eagle grabbing a snake in its talons. Therefore, this image is on the Mexican flag because it represents the birth of the Aztecs and much of Mexico's Cultural Institutions.