Why was it the Europeans in 1400–1600 CE who explored and then eventually imposed their will on the world by 1850?

The answer to this varies when we look at different world regions. Let's start by looking at the Americas.
It really is notable that Europeans were able to dominate the Americas in such a short amount of time after first arriving there at the end of the fifteenth century. This is mostly due to the extreme population crash of the natives that happened soon after the Europeans' arrival. It is estimated that there were well over one hundred million inhabitants of in the Americas before the Europeans arrived. As a result of Old World diseases that Native Americans had no immunities to, the population quickly crashed by as much as ninety percent in just a few decades after contact. This left a weakened and disorganized population that really stood little chance of resisting European conquerors.
Many Europeans saw the Americas as a divine gift left for them. They did not fully understand the role that their diseases played in presenting them with vast regions ripe for colonization and exploitation. Many saw it as their destiny to extend their will and sovereignty over these continents. Why else would God have presented them with such a wealthy land almost entirely devoid of people who could resist them?
The colonies that nations such as Spain, Portugal, England, and France built in the Americas made them incredibly wealthy and further fueled the notion of European superiority and the desire to build empires and extend their influence around the world. They then turned their attention to Asia.
The people of Asia were not susceptible to the same diseases that the people of the Americas were. Their population usually rivaled that of Europeans in size and strength. Consequently, Europeans did not outright colonize the continent. However, a huge demand for Asian goods fueled European interests in the region. Aside from the British presence in India and the Spanish in the Philippians (in addition to small Dutch and French trading colonies in the Pacific), little conquest was made prior to 1850. However, a strong European economy allowed European merchants to exert their control over trade routes and certain luxury markets in Asia beginning in the 1500s.
Africa was a different matter. Few Europeans had been to the interior of the continent before the latter part of the nineteenth century. However, the slave trade greatly affected many parts of Africa. The need for laborers in the New World after the population crash and genocide there compelled many Europeans to trade for slaves along parts of the African coast to have a reliable supply of labor. They took advantage of inter-tribal conflicts and the greed and ambition of some African rulers to supply them with slaves. This had a profound effect on certain parts of the African continent as warfare increased and whole tribes were decimated.