The large number of countries can be attributed to what colonial powers occupied the land, how the colonies were set up to be governed and how the colonies gained their independence.

South America was basically split in half between the Portuguese and the Spanish. Portugal had the massive land holding that was Brazil and the Spanish had basically the land surrounding Brazil to the West. Portugal operated the Brazilian colony in a way that did not separate the state into smaller jurisdictions, at least not in a way that caused any different cultural identities to appear. Spain, on the other hand, operated their colonies in a much more regimented way splitting land holdings up between different regions.

Spain’s colonies operated under four levels of administrative hierarchy. Each of these levels covered a certain area and split these areas up and governed them according to rules specifically for that region. Over time, this caused differences to arise in the way that the citizens expected to be governed and thus eventually created semi-autonomous cultures to appear.

Within South America, Spain had three different viceroys (the highest level of administrative government in Spanish Colonial government) New Grenada (Colombia, Ecudor plus some Central America), Peru and Rio de la Plata (Argentina, Bolivia, Uruguay, Paraguay). Each of these viceroys had separate governing bodies as well as their own ruling classes. Because of this, there were noticeable differences in the people and government in these viceroys by the time they gained their independence.

Spanish colonies in South America gained their independence from 1810–1830 due to Napoleon’s conquest of Spain. This conquest led to legitamacy issues in the colonial governments which eventually led to declarations of independence. Since the colonies were gaining independence at various times and not all together, the governments were being set up along administrative borders and following administrative procedures.

South America borders lie as they do currently due to the separate administrative regions of the Spanish colonial empire gaining independence at different times and forming their own governments based on smaller administrative borders that previously made up the colonies as well as forming a country that could connect culturally. There was a man, Simon Bolivar, who sought to unite South America as a United State of South America but that is a story for a different time.

There's definitely more room for discussion here, but we do have a whole subsection in the FAQ on this topic that you might find interesting. Answers of special interesting for this one by flaired users would be:

• Why is there not one united nation across all of Latin America? by /u/LegendaryTubaHero

• Why did North America split into 3 countries and South America split into 15 though they were colonized similarly by /u/Yodatsracist

• During the early 1800s as Spanish holdings in South America began to vie for independence, was there any concerted effort to create a "United States of South America"? If so, why did it fail? by /u/ThucydidesWasAwesome

As a very short version: One the one hand the post-independence fragmentation often followed colonial organizational units that were already made up of more or less seperate parts that could only trade with Spain and not among themselves (e.g. the larger viceroyalties, but also smaller units like the audiencias, capitanias, depending on each region); on the other hand, further fissures go back to the earlier independence period and the conflicting interestes of different independence leaders (like the major leaders San Martín and Bolívar, as well as some of the latters' generals) and other, traditional creole elites. There were even plans in the early 1800s of forming a pan-Latin American confederacy that did not work out for such reasons.

All this can be strongly contrasted with Brazil, which in the same period and also because of Napoleon's Iberian invasion actually had the Portuguese court move there, meaning a much longer royal/imperial, centralizing influence - whereas in Spanish America the Napoleon invasion and his capture of the Spanish king was the starting point for the drawn-out wars of indepence.

But I'd really recommend to read these great anwers above for more details on these questions.

this change could be attributed to war itself. whereas the independence process was long and bloody for Spanish America it was rather peaceful in Brazil.

even though the different American factions fought the royalists together, when it came to the incipient politics of the newly independent countries they gad forged, division was inevitable. even within national congresses, division among the ruling elites meant that spanish America was fragmented.

Brazil's independece came as a decree of the Portuguese crown who was already living in rio. no war of independence and a much smaller region to rule. the elites were certainly divided but not to the point of creating different independent states (uruguay was the exception)