Why did people believe that the earth was flat?

No one believed the Earth was flat (beyond present day "Flat Earthers"). This is a myth that has been perpetrated in education because of a mistake made by an early 20th century text book author. Many people were aware that the Earth was a sphere by the 3rd century B.C. Both Claudius Ptolemy and Marinus of Tyre (early geographers) proved that the Earth is a sphere. Working from data compiled centuries before, their calculations, while still inaccurate, where a great improvement on earlier work.

Christopher Columbus did not set out to prove that the world was round, but rather set out to find a faster trade route to Asia, discovering that the Earth was larger than anyone originally believed and discovering North America on behalf of European civilization. North America had been discovered by other populations before then, including the Vikings and of course the Jomon (ancestors of the Japanese), but Columbus' discovery was probably more relevant to modern civilization.