First, let's be clear: He did not "discover" that the world was round. This, and the world's approximate size, were reasonably well-known since the 2nd Century BC. Sailors of the age (late Medieval) knew that there were riches to be gotten n China and India, but the journey around the horn of Africa was prohibitively long. Going West, around the globe, seemed like a bad idea, as no ship was large enough to stock the food and water needed for such a long journey.
Columbus tried it anyway. He thought Asia was a lot closer than others believed it was. When he found the Bahamas and the islands of the Caribbean, he thought he was in Asia. He didn't bring much gold back to Spain, but he did bring back evidence of a new, unknown continent. Instead of a months-long journey that would starve and kill any crew foolish enough to take it, Columbus' first trip across the Atlantic took a little more than a month. Other captains, knowing that a trip across the ocean could be done in that short a time, took the voyage without hesitation.
The Americas were ready for plundering, and many ships (mostly from Spain at first) set out to loot the Americas for gold and bring it back to Spain. This created a new industry, Caribbean piracy, which focused on attacking and looting Spanish galleons.
Suddenly the world was smaller and its contents and dimensions were well known enough for European explorers to set out for all parts of it. Areas that had never encountered European sailors, or that had avoided frequent contact with them, were suddenly besieged for trade, plunder and colonization.