The long expansion of European power from Europe itself to the rest of the world began in the fifteenth century. One of the great world-historical conundrums is why European states expanded so rapidly and aggressively, while other powers like that of China, the Ottoman Empire, or the Indian kingdoms did not. Why was it Europe that took over the Americas (and, much later, much of the rest of the world) rather than Persia, the Ottoman Empire, India, or China?
Ironically, one of the most likely answers to that question is that it was Europe’s relative poverty as compared to the states of the Middle East and Asia that led Europeans to seek out new sources of wealth. Whereas the intra-Asian trade routes linking China, Korea, Japan, the islands of the western Pacific, Southeast Asia, and India ensured that Asian states enjoyed access to wealth and luxury goods, Europeans had to rely on the hugely expensive long-distance trade between Asia, the Middle East, and Europe to access goods like spices and porcelain that Europeans desperately wanted (so we can conclude based on the prices elite Europeans were willing to pay for them) but could not produce themselves.
The demand for trade with the east was limitless in European society. Luxury goods from south and east Asia were always among the most sought-after commodities in Europe, stretching all the way back to Roman times. Spices were worth far more than their weight in gold, and Chinese goods like porcelain and silk were also highly prized. Enterprising merchants who were able to position themselves somewhere along the Indian Ocean trade routes or the famous Silk Road between Europe and China stood to make a fortune, but the distances covered were so vast that it was very difficult and perilous to take part in mercantile ventures. Thus, Isabella of Spain was not alone in funding explorers who sought to reach the east via easier routes when she hired Columbus.
The situation became even more difficult for Europeans thanks to the rise of the Ottoman Empire, as well. When Constantinople fell to the Ottomans in 1453, the traditional trade routes to Asia were disrupted, particularly as the Turks started taking over the Venetian maritime empire. Likewise, Europeans had long traded with Muslim merchants in North Africa for gold, ivory, and spices, and they longed to cut out the middlemen and get to the sources farther south. Some of this was doubtless born of anti-Muslim prejudice, but there was also the simple fact that the Ottomans now directly controlled a major link in the East - West trade axis, deriving profits that Europeans desired for their own.
In addition, the crusading tradition, especially that inspired by the Reconquest of Spain and Portugal, served as an inspiration for European explorers. The Reconquest was only completed in 1492, the same year that Columbus sailed in search of a western route to Asia, and many of the Spanish conquistadors (conquerors) who invaded South and Central America afterwards had acquired their military experience from what they considered to be the holy wars against the Muslim inhabitants of the Iberian peninsula. That crusading ideology was easily adapted for the purposes of conquering vast American territories and forcibly converting the Native American inhabitants to Christianity.
There were thus economic and cultural reasons that Europeans wanted to reach African and Asian commodities and wealth. They were able to access that wealth thanks to technological advances. Until about 1400, Europeans had no ships capable of sailing across an entire ocean (the Viking longboats of the Middle Ages were an exception, but they were no longer in use by the Renaissance era), and the European understanding of geography and navigation was extremely primitive. From about 1420 on, however, maritime technology improved dramatically and it became feasible to launch voyages that could cross the entire Atlantic Ocean with a reasonable degree of certainty that they would succeed. The key here was the invention of the caravel, a new kind of ship that was able to sail both with the wind and against lateral winds; as long as the wind was not blowing in the opposite direction one wanted to travel in, it was possible to keep moving in the right direction. Reasonably effective compasses and a device to measure latitude called the astrolabe came into European hands from the Middle East around 1400 as well. Thus, by 1400 Europeans had both a number of reasons to want to explore and for the first time had the technological means to do so.
Despite those advances, the European grasp of geography remained very shaky. As of 1400, Europeans had terribly imprecise knowledge about the rest of the world. They did not, of course, know anything about the Americas. They tended to confuse “India,” “Cathay,” and “Japan” with “Asia” itself. They had a vague notion that all of Asia was ruled by Khans, in part because of the popularity of the Venetian merchant Marco Polo’s famous account of his travels undertaken in the latter part of the thirteenth century. Polo was a Venetian merchant who had traveled to the court of the Mongol Khan Kublai and eventually returned to Europe, but his account merely reinforced just how far away, and difficult to reach, Asia was taking the usual eastern routes. Many sincerely believed that monsters occupied the interiors of Africa and Asia, and besides Polo, no Europeans had ever made the trek to the far east and returned to tell the tale.