The sentence that correctly uses the term "New World" is C: Fifteenth and sixteenth century Europeans saw the Americans in the New World. This is as this sentence clearly outlines the distinction at that time between Europe and the Americas (and to some extent Africa and Asia). It must be understood that at that time, there was no way for persons in one continent to contact individuals in another continent; in fact, the existence of the continent of the Americas was quite unknown. It was only the calculations and assumptions of a few navigators in Europe that gave any indication that such a place existed.
So once, Christopher Columbus and other navigators after him proved that the Americas existed, it became a "New World" for the Europeans. It was more accurate to say that it was an exotic and different environment for the Europeans than 'new' as there were individuals living there at the time of its 'discovery' by the Europeans and societies that had been there for centuries.
Response A means that Vasco de Gama brought the European culture to India which was quite new and different for the people there. While the meaning is similar to C, the correct use of "New World" is strictly applicable to the Americas as the Europeans had no contact with that part of the world prior to 1492.
Response B means that the exploration brought unknown facts to the attention of the Europeans which they added to their existing knowledge.
Response C simply means that that was the terminology Columbus used for his navigational ambitions.
