There are generally said to be three main factors that motivated Europeans to explore and colonize the New World. These are given as “God, gold, and glory.” Of these, the last two are typically believed to have been the more important.
Europeans wanted to explore and colonize for “gold.” That is, they wanted to get rich by finding new lands and exploiting their resources. Individuals wanted to get rich and so did the countries that sent the individuals out to explore.
Europeans also wanted glory. They wanted to be personally famous and to be given titles that would make them part of the elite of their society. Countries also wanted glory. They wanted the prestige of having large empires.
The desire to bring Christianity (“God”) also played a role in this process. However, historians generally agree that the more worldly motivations played a greater role.