I can't really tell you about how it went on in Argentina and Chile in great detail, but as for Brazil, there were two main factors that made it an attractive country for the fleeing nazis. The short answer is: a vast and sparsely inhabitated territory coupled with collaborationist help created a favorable scenario for nazis who wanted to escape justice.
For more detail, Brazil is a vast, and to this day still very empty country in many areas. It was even more so in the latter half of the 1940s. Even if there were already some very large metropolitan areas then, the majority of the population lived in rural areas or small towns close to them, and the size of the territory made it hard for government action or even infrastructure to cover the entire country evenly. This made it easy for wanted criminals to slip by if they were willing to hide in a small village away from easy reach of government attention, mass media, paved roads and other such things that could contribute for them to be found, identified and have their whereabouts reported by people who would care enough to do it.
Being in a foreign country, between people who had no way to know their previous identities¹ or link them to it made for an anonimity that partly accounts for how the german "immigrants" who arrived in the last stretches of the war and just after its end found a modicum of security in their destinations. The history of german immigration to Brazil and the large germanic population in the south of the country also made sure the fugitives didn't stand out.
This also ties in to another important element to explain why the nazis chose Brazil as a destination to evade capture. Between 1928-38, NSDAP's Landesgruppe Brasilien was the biggest nazi party outside of Germany in terms of absolute number of affiliates² - in its prime, the numbers were around some 3000-5000 people who were either affiliated to it or engaged in party activities. This is directly traceable to the considerable influx of german immigrants in the nineteenth and twentieth centuries to work the fields in the brazilian south after the slaves were freed. By the time the NSDAP rose to prominence in Germany, german 'colonies' were effectively stablished in the country, and the immigrants and their descendants played the balancing act between integrating and preserving their original traditions and culture.
Thus, with the rise of fascist ideologies along the world in the decades after WWI, some brazilians, mostly german descendant members of the elites, began organizing nazi political groups - some joined as an effort to cultivate what they perceived to be a link their family's mother country, some out of peer pressure, and still others due to genuine ideological conviction, as the nazi doctrine aligned itself pretty well with how racism manifested itself in Brazil.
Eventually, the Landesgruppe Brasilien was officially formed and registered. Even if it didn't ever have any sort of significant influence over brazilian sociocultural or political life in the grand scale and is understood today to have been rather decentralized and poorly organized for most of its existence, the fact that it helped officially spread their ideology in the southern and southeastern states explains the reason why the nazis found receptive people to turn for help later on.
This relative lack of significant political influence on the federal level also helped the party to grow without facing strong opposition. Up until 1942, Brazil was neutral in the conflict and there was no widespread public condemnation of fascism in the country. This policy of tolerance was incentivized by then-dictator Getulio Vargas', who conducted a pragmatical international policy consisting of on one hand suggesting siding with the Axis, while on the other courting the US, testing the waters to better decide which side's proposal for an alliance brought the most benefits to the country. Before Brazil joined the war, though, in an effort to stop the spread of international influence in brazilian politics, Vargas published a decree that outlawed all political parties affiliated with or tied to foreign institutions - such as the NSDAP.
In april 18, 1938, the brazilian nazi party was declared illegal and disbanded. However, that didn't stop the former members from keeping their ideological convictions. This allowed for the sympathizers to become active collaborationists when called upon by the network of contacts they kept within Germany. Their assistance ranged from providing falsified documents and visas, transportation, lodging and eventually even permanent accomodation, preferably somewhere secluded and discreet, such as farmland properties far away from major population centers.
The combination of the two previously mentioned factors - the collaborationists willingness to help the escapees and the possibility of hiding away in remote, but comfortable places where they could reasonably expect to live the rest of their days in peace - explain why the nazis favored Brazil as a destination to escape pursuit. In many cases the remoteness of the chosen refuges allowed many nazis to keep a low profile and slip by unnoticed for decades, while in other situations they would also have complicit local law enforcement and government agents knowingly cover them up and not report them to higher authorities.
It was known by the international intelligence community that the three South American nations mentioned in the question were preferred escapee destinations, and Mossad operations were able to find targets who hid in Brazil for decades after the war, such as Herbert Cukurs, who was assassinated in 1965 - but even a fugitive as notorious as Josef Mengele, who drowned in a beach in 1979, lived for the rest of his life in brazilian soil incognito and was only posthmously identified.
As for Argentina and Chile, the only comment I feel confident enough to make about the matter is that there was also a considerable number of sympathizers-turned-collaborationists, which resulted in an attraction factor for escapees similar to what was present in Brazil, as far as the possibilities of receiving local help were concerned.
¹ - Or this was the expectation. In the cases when suspects were found and detained for questioning, holocaust survivors were called in when possible to help determine the identity of the suspect. This was the case of Stanislaw Szmajzner, for instance, who was intimated by the police to identify SS officials Franz Paul Stangl in 1967 and Gustav Wagner in 1978.
² - It's worth noting that Chile and Argentina had what ammounted to fewer total affiliates in their respective nazi parties, but due to much smaller populations in comparison to Brazil's, this translated as a higher percentage of people engaged with nazi political groups in the former two countries when compared to the latter.
Bibliography:
"Ein Volk, Ein Reich, Ein Führer! A seção brasileira do Partido Nazista e a questão nacional.", by Luís Edmundo de Souza Moraes.
"Fascismo no Sul do Brasil", by René Gertz.
"Nazistas Entre Nós: a trajetória dos oficiais de Hitler depois da guerra.", by Marcos Guterman.
Edited for better formatting and to add information and sources.