Why were European countries interested in exploring the New World?

There are several reasons European nations wanted to explore the New World, but for the most part they can be summed up by economic motivations. 

Exploration of the seas and eventually the New World (the North and South Americas) was initially an attempt to find a sea route for trade with Asia. Asian countries offered valuable goods that Europeans wanted to have access to like spices, silk, and precious metals. Transporting these goods over land was labor intensive and often dangerous as it required long trips through deserts and mountains. A sea route would be both quicker and easier for transporting goods back and forth. 

After Europeans traveled to the New World, they saw much potential for economic (and possibly spiritual) gain. The Americas had a number of animals and resources Europeans hadn't encountered before, and these became very hot commodities on the continent. Colonies were established to exploit the land and resources of the Americas with a secondary intention of Westernizing the native people. Westernization included conversion to Christianity, learning to speak one of the colonial languages, and taking up European customs. Many Europeans felt that it was their duty to Westernize Indigenous Americans because their traditional cultures were savagery and that they lived unhappy lives and would not be safe in the afterlife. 

Europeans also had a sense of wonder and mystery at the unknown lands of the Americas and believed that there may be supernatural places or beings waiting to be discovered. Most famously, many searched for a Fountain of Youth in the Americas, but to no avail. However, some plant, animal, and mineral materials were believed to have special qualities for health and were transported back to Europe for use as medicine.