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 fifteenth century the Turkish Empire had gained control of those overland routes. They charged high tariffs for passage through...

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 fifteenth century the Turkish Empire had gained control 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 countries also 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.