So far as we can tell (given the lack of written records) the Aztecs were able to create their empire because of their ruthlessness and their ingenuity.
The Aztecs migrated down from the North into what is now called the Valley of Mexico. They were a migratory tribe which came into contact with the sedentary tribes of the valley. The tribes of the valley used them as mercenaries because of their warlike nature.
The Aztecs were brutal enough (they asked a ruler for his daughter to be their queen and then sacrificed her) to end up being chased out by the more powerful tribes. They then settled in marshy areas around the lakes where no one would bother them. They built this area up (this is where their ingenuity came in) by doing things like dredging the lake bottom to create more land.
As they did this, they became more prosperous and powerful and were able to use this prosperity (and their ruthlessness) to conquer the surrounding area militarily. As time went by, they enlarged their empire through use of military force. In this way, the Aztecs created their empire largely through their military power and ruthlessness.