Why did Europeans, and not Asians, undertake the voyages of discovery connecting Old World with New World?
The answer is quite simply that the Asians had goods Europeans wanted very badly, and not vice versa. The Europeans were especially hungry for the spices the Asians could cultivate and they could not, such as pepper and cinnamon. In earlier years, the Europeans had made the overland journey to the Far East, but by the 15th century the Turkish Empire had gained controlled of those overland routes. They charged high tariffs for passage through their lands, and the Europeans also ran a higher risk of encountering pirates and bandits. An already costly overland journey became prohibitively expensive, taking much of the profit out of the spice trade.
Countries such as Portugal had been developing their navigational technology and improving their shipbuilding techniques. They wished to find a sea route to India by sailing around the tip of Africa, so as to circumvent dealing with the Turks. When they were able to successfully do this, it unleashed a frenzy of exploring activity as other also countries sought alternative overseas routes, seeking profits in trade with Asia.
In the process of trying to find ways to Asia, the Europeans stumbled on the North and South American continents, and almost immediately realized the huge potential existing there in raw materials. This created competition for the New World's resources among European nations and thus more exploration.
Considering that much of the technology that permitted Europeans to traverse the oceans came from Asia and the Far East, the reason is not found in any technical advantage that Europe possessed over Asia. In answering this question, it is important to understand the motive for European exploration and discovery of America. First, the Europeans wanted an easier and less expensive route to the East in order to acquire spices, drugs, perfumes, and silks that were in very high demand on their continent after the Crusades. So through this perspective, the Asians would not be trying to find a sea route to their own lands. Secondly, the Europeans were in the midst of the Renaissance, a period of time in which there was a belief in the unbelievable potential of humanity. This period brought about scientific discovery and reason. Despite the fact that Europeans discovered a previously unknown world, this was done by complete accident and for years following the landfall of Columbus, Europeans still did not realize they were not in Asia.
There were reasons why the Europeans and not the Asians undertook the voyages to connect the Old World with the New World. One reason had to do with safety. There were times when the land routes to Asia were either blocked or unsafe due to possible attacks. Islamic Empires were often fighting for control of the areas where the land routes went. Also, robbery was common as the caravans were often in danger of being raided.
Another factor dealt with the growing competition between the European powers. The Italian city-states were in a good location to profit from and dominate the trade with Asia. Spain, Portugal, and Great Britain hoped to break the Italian control by finding different water routes to Asia.
Finally, the Europeans were more interested in Asian products than the Asians were interested in European products. Silk and spices from Asia were highly demanded in Europe. Thus, the Europeans wanted to find multiple ways to get these products to Europe.