The correct answer to this open question is the following.
European Empires expanded into the Americas through navigation and exploration. Indeed, part of the Middle Ages is considered to be the Age of Exploration. A time in which European kings ordered the navigation and exploration of distant territories to discover new trade routes to the West Indies and Africa. That is how they discovered teh Americas. As was the case of Christopher Columbus, who discovered America for the Spanish crown on October 12, 1492. In those years, there were also other notable explorers such as Ferdinand de Magellan.
European conquerors and colonists had many advantages over the Native American Indians and Mesoamerican Indians. They had horses that impressed the Indians. They had armors that impressed the Indians. ANd they had fire weapons and gunpowder that impressed and terrified the Indians. Furthermore, Europeans brought to the Americas diseases such as malaria, cholera, influenza, or chickenpox, that almost wiped out the population of Native Americans.
And the main motive of European conquerors: big profits by exploiting the many raw materials and natural resources of the Americas. And yes, the evangelization of the Indians into the Christian religion.