Why were kings so desperate and determined to pay explorers so highly for a new route to East?

Because they weren't able to trade with East through the route that was established in the Middle Ages.

Explanation:

After the Ottomans conquered the Byzantine Empire they were the ones that dominated the route that connected Asia and Europe. Traders were not safe on that route and that is why rulers of European countries decided to finance sailors to find a new sea route. The first ones that did that were rulers of Spain and Portugal. Others followed.