There were many factors which led to the European exploration of other lands.
The Renaissance period increased people's curiosity about the world. They became interested in the idea of adventure and discovery which encouraged the exploration of other lands.
European states were engaging in war and thus rulers required resources and precious stones like gold to pay for the wars. As such, exploration was undertaken to find these resources. In addition, exploration was undertaken so that they could colonized states and bring back slaves.
It was a period when nation states were rising. Therefore, exploration was undertaken because states want to expand their empire for more power.
The defeat of the Spanish Armada was a turning point in history. Spain had the first actual permanent settlement in the New World before the English. Spain was out for gold, and not really fond of colonizing. But with the defeat of their navy, this leads other nations to take a look at making settlements in the New World.