Question

Why were Europeans eagerly searching for sea route to Asia in the 15th century?

Answer 1

Assuming you are looking for a single answer, the best answer is:

They needed to find an alternative to the silk road due to the rise of the Ottoman Empire.

The Ottoman Turks were involved in trade and merchant traffic themselves, and they often charged high rates for travel through the territories they now controlled.  Also, as religious and cultural opposites, European Christian culture and Islamic culture often came into conflict. So finding an alternate route was appealing to Europeans who sponsored exploration by sea.

Curiosity was also a factor, but they were not so much about "seeking out new live and new civilizations."  If anything, they wanted to find further possibilities for extending their own civilization and profiting from trade with others.

To "boldly go where no one has gone before" is the theme for exploring the universe set forth in the Star Trek television and film series.