The French explore and colonized the part of the North America in order to look for gold and trap for furs. The French initially came to the United States as explorers, who were trying to find a route to the Ocean and resources.
The French finally attempted to build some colonies throughout North America that abandoned, due to various reasons such as disease, weather or struggle with other European authorities.
Though during the course of time, they established colonies in the Americas, their initial motive was to look for gold and trap for furs.