Sorry for the lateness in the response, as I have been kind of busy as of lately.

Columbus arrived to Queen Isabella and King Ferdinand after having talked to a lot of people, a rather heavy lot of important people. So, first we have to get to explaining how did he manage to do that. Christopher Columbus arrives in Castile about 1484-5, to the region of Huelva. Why did he go there? He did have some in-laws, most particularly Miguel Muliart, who was married to Briolanja Moniz, who was Felipa Moniz's sister. Felipa Moniz was Columbus' wife, and mother of his son Diego Colón Moniz, who would become the II Admiral of the Indies.

The region of Huelva is a very important maritime zone, with notable ports, particularly at that moment. From Palos was the Pinzón family (Martín Alonso, Francisco Martín, and Vicente Yáñez), known as some sort of natural leaders of the zone being amongst the richest of the land. The next village is Moguer, where the leading family was the Niño lineage, a family of shipwrights. Close to Palos, there is the monastery of La Rábida, then inhabited by two good cosmographers, fray Juan Pérez and fray Antonio de Marchena. Last but not least, the lordship of Palos was linked to the Dukes of Medina Sidonia. Another notorious resident of Palos was Pedro Vázquez or Pedro Velasco, who had experience navigating the Atlantic, as he had taken part in the expedition of Diogo de Teive who discovered the island of Flores.

With this context in mind, Columbus was lucky enough to get to talk to Juan Pérez, who had been related to the Royal Court and who had important acquaintances. This led to Columbus being introduced to the duke of Medina Sidonia, who took a liking to Christopher Columbus, considering him a very clever man. Medina Sidonia introduced Columbus to Pedro González de Mendoza, the Great Cardinal of Spain, a man so powerful that he was called "the third king of Spain". Mendoza took Columbus under his wing and had him introduced to Alonso de Quintanilla, one of the thop dogs at Court. Quintanilla had been very active during the war of Castilian succession unofficially sponsoring all sorts of corsair activities against Portugal, something he kept doing for the years to come, employing seafarers from Palos, Huelva, and Moguer. There are many records of pirate activities by people from Palos between 1477 to 1485, attacking Portuguese ships around the Canary islands, the cape of Aguer, Guinea, and as far as the Azores.

Between 1486 and 1492, Columbus kept following the Royal Court, basically leaving off the mercy of Quintanilla and Mendoza, until the war with Granada could be concluded. In 1491, as attested by Galíndez de Carvajal, Columbus had an interview with the Catholic Monarchs, but the Monarchs submitted Columbus' plan to a comittee of experts from Salamanca, who deemed Columbus' plan foolhardy and impossible, as Columbus' calculations of the Earth's size were 10,000 Km short of the actual size.

This leads to the main question as to how did Columbus eventually convince the Monarchs, something he did not quite do himself on his own. In 1491, in the camp of Santa Fe, Columbus eventually grew tired of waiting to have his paperwork sorted out, so he got his stuff and left, deciding to try his luck in France. Luis de Santángel, treasurer of the Holy Brotherhood and chief accountant of Aragon, sent a messenger to get him back, as he secured the funds for the expedition, taking 1,140,000 maravedis out of the treasury of the Holy Brotherhood on April the 2nd, and getting his money back from Hernando de Talavera to be paid with funds of the Bull of the Holy Crusade from the Bishopric of Badajoz.

This, still leaves the question unanswered, and we can only speculate as how Columbus convinced Isabella and Ferdinand, but the speculation seems consistent: Alonso de Quintanilla thought that the expedition would be worth it as the cost was not really high, and if it succeded, they could completely sabotage Portugal's Atlantic policy and Portugal's trade with the Spice Islands. Mendoza backed Columbus, Quintanilla too, Santángel secured the funding, and the ships would be basically free as the Queen issued a letter to Diego Rodríguez Prieto and other neighbours of Palos telling them to give two caravels for Columbus to use, as Prieto and others had a debt towards the Crown.

Sources

FERNÁNDEZ DE OVIEDO, Gonzalo (1535), Historia General y Natural de las Indias. Seville: Cromberger.

CASAS, fray Bartolomé de las (1559), Historia general de las Indias. Manuscript in the BNE, digitised here.

FERNÁNDEZ DURO, Cesáreo, ed. (1892). Colección de documentos inéditos relativos al descubrimiento, conquista y organización de las antiguas posesiones españolas de ultramar, 2ª serie, tomo 7 y tomo 8. Madrid: Real Academia de la Historia. Volume 7, linked here. Volume 8, linked here.

GARCÍA CRUZADO, Eduardo, coordinator (2015), Actas de las Jornadas de Historia sobre el Descubrimiento de América. Palos de la Frontera: Universidad Internacional de Andalucía, and Ayuntamiento de Palos de la Frontera.

ANDRÉS MARTÍN, Melquíades (1967), El dinero de los Reyes Católicos para el descubrimiento de América, financiado por la diócesis de Badajoz. Badajoz: Editorial Cisneros. Digitised by the Library of Extremadura.

FERNÁNDEZ-ARMESTO, Felipe (2004). Cristóbal Colón. Madrid: ABC.