Why did kings sponsor voyages to explore the indies?

Because the lands in the West Indies had many resources to exploit that disappeared from Europe after hundreds of years of this process. Many nations wish to have these resources to gain an advantage of a rival nation. The West Indies was capable of growing coffee an sugar, something that Europe couldn't grow because of it's cold climate. So if European empires took the West Indies to exploit these rare resources an gain an economic advantage over other nations.

The other answers are ( mostly) all correct but what they don't emphasize is that in a day before refrigeration and the only way to preserve meet was to salt it, spices were literally worth their weight in gold. Think about that the next time you are in the supermarket....ten packages of cloves would make you solidly middle class in 16th Century Italy . The spice section in any supermarket would be more than most cities would ever have hope of seeing.

For centuries the trade of valuable commodities sush as silk and spice had been controlled by merchants in the Middle East who had charged exorbitant prices for their merchandise. By findin anouther route that was not controlled by these traders the Europeans sought a way to buy these goods directly from the producers and at considerably less cost.

The king would also benefit from the riches of trade and the Indies was a very popular trading port back then.

Promote the spice trade and Christianize the natives. A water route was to be developed rather than the overland route. The overland route was fraught with dangers; such as bandits.

They wanted to directly access the spices instead of through middle men in the Middle East, paying exorbitantly, through their noses.

To find a trade route. To make money.

Money, trade makes money.

They had all the money.

GREED