Historically, it is not completely clear why the Aztecs chose the place that they did for their home. Historians believe that they picked the place they did because they had been chased out of all the better places. The Aztec legends, by contrast, say that they picked the place because they were given a sign by their gods.
According to the legend, the Aztecs saw an eagle sitting on a cactus on a small island in Lake Texcoco. The eagle was clutching a snake in one of its feet. The Aztecs took this as a sign that this was the place that was meant for them. This legend is the origin of the design of the Mexican flag today.
Historians take a less romantic view of things. They say the Aztecs were a group of nomads from the north who wandered into the Valley of Mexico, where there were much more civilized, sedentary groups. The Aztecs proceeded to antagonize the people of those groups through their aggressive and warlike ways. The people of the valley then drove the Aztecs away, forcing them into the swamps around Lake Texcoco.
The Aztecs were once told by their god, Huitzilopochtli, that they would find their home wandering around and then see an eagle sitting on a cactus eating a snake.And it so happened that they did