The Ottoman Empire controlled the Silk Road, which, at the time, was the only trade route that connected Western and Eastern Europe.
And, of course, Europeans were demanding silk and spices, and the rulers wanted to please their subjects.
When you think about it, it made sense for Europeans to search for a sea route. They knew nothing about the unexplored portion of the globe, and their imaginations ran wild. There were people who thought that there were cities made of gold and fountains that could reverse aging.
So, a sea route to Asia? Of course there was a sea route to Asia, they said.