Waldseemüller studied reports by Christopher Columbus and Amerigo Vespucci and decided Columbus was wrong in thinking he had reached India. Waldseemüller favored Vespucci's theory that the lands they were exploring on the other side of the Atlantic were actually part of a previously unknown continent. Waldseemüller rewarded Vespucci by naming the continent after him. America is the feminized Latin form of Vespucci's first name. All other continents had Latin feminine names, so it fit.
The map is not only correct about the New World, but also portrays other parts of the globe far more accurately than other maps of the time. It's a fine work of art too, with detailed depictions of terrain and portraits of Ptolemy and Vespucci.The map is on display as part of the exhibition "Exploring the Early Americas."
Thank you, Gadling ;)