The most direct answer to the question of exactly how the world was eventually fully mapped would be to say it was done piece by piece.
The earliest attempts to map the “world” were done by those who could only assume that what they could see and what they were aware of comprised the entire “world”.
As we would come to realize later in history what they thought of as the world was their world which of course was not the entire world!
Broadening the Horizons
As a civilization undertook exploration to the farthest corners of their known worlds they often encountered other civilizations that had knowledge of their own slice of the world pie which was then incorporated into the big picture making the world bigger and bigger with each new entry.
Early exploring civilizations that were seafaring would often encounter places that had never thought to have existed especially when you consider that a good portion of the populace still thought that the world was flat and that you would fall off the edge if you sailed to the horizon.
The First World Maps
The very 1st maps of what the world was thought to look like came from ancient Greece with academics such as Anaximander in the period 610-546 BCE comprising a map of the lands grouped around the Aegean Sea being at the center with all-encompassing oceans surrounding those lands. A rudimentary attempt to be sure, but with the information at hand the best that could be expected.
There were many other attempts over the millennia that would expand that horizon but with still incomplete information being the core.
The world’s first globe appeared in 1492 and was introduced by Martin Behaim but as Columbus had not yet returned from the New World the New World or North America was still a mystery and was represented by empty ocean at the time.