Columbus tried for years to convince European monarchs that a trade route could be found to Asia if they went west. All expeditions went east, never west. The biggest argument for the educated was not that the earth was flat, but the distance was too far.
The common slobs of the day were always told that the earth was flat. Eventually there is a edge, and after that edge, the abyss. The Catholic Church told them that the earth was flat, thinking otherwise was blasphemy. Blasphemy was dealt with by torture and death, usually very harsh in both acts.
For the educated, they had known since the ancient Greeks and Egyptians that the earth was round, the big question was how big? Simple geometry and astronomy led the Greeks and the Egyptians believe they had the answer. Both answers however were made the circumference of the earth too, too big. This meant starvation and death for any crew trying to sail.
A simple solution would be to make bigger boats, but that was not possible. Technology at the time only made it possible for ships to have 3 masts. Why not bigger ships with four masts? The stress on the frame of the ships would split it apart. The size of sails and the types of wind that a ship would experience in the open sea would not make it prudent to make ships with 4 masts.
Therefore, the continent of Europe was stuck with ships with only 3 masts. The difficult pitch Columbus had to face was finding someone willing to almost throw away ships, materials, and men. Also, Columbus was not a nice man and did not sell his plan very well. He was laughed out of every country in Europe.
In the end Spain ruled by Queen Isabella agreed to finance this suicide trip. Why? Spain had some serious problems.
Spain was the laughing stock of Europe. They were the bottom of the barrel when it came to power. Being the country furthest west made trade with the Middle East next to difficult. They were poor and lacking many things found in other countries. If Columbus was right, then they would move up the ladder of power in Europe.
Spain just came out of a difficult civil war removing the Muslims from their country. The war was costly and many people were upset about what they lost. Again, if Columbus could actually do what he said, then they would now be the closest to Asia and become the richest.
Queen Isabella was a woman. Not many monarchs respected women. If Columbus did succeed, she would personally gain respect.
Lastly, with the amount of debt that Spain was under, sacrificing three small ships, a crew, and materials was not a big loss. It was understood that Columbus had to advertise exactly what he was doing when finding a crew. The kings and queen believed that this would be the end of the mission; however more people volunteered than spaces on the ship. Columbus was ready to sail.