Prior to WWII, Argentina was the 7th most wealthy nation in per capita income ahead of Italy, Spain, Denmark, Japan, and Brazil. Only Switzerland, New Zealand, Australia, the United States, the United Kingdom and Belgium were wealthier in per capita income.

Argentina was an attractive economic destination for many immigrants including Germans. Argentina experienced a massive wave of European immigration second only to the United States in the late 19th and early 20th century. Furthermore, Argentina renewed ties to Spain in the 1930s. Germany and Spain had Fascist governments with relatively close political ties just prior to WWII, and both Spain and Germany cultivated good relations with Fascist leaning Argentina. Argentina pointedly stayed neutral during WWII in the same sense that Spain stayed "neutral"/Axis sympathetic. Argentina only declared war on the Axis Powers a month before the end of World War II in Europe when the writing was on the wall so to speak.

As a prosperous Axis sympathetic nation with an existing large German immigrant population, Argentina was a natural refuge for displaced Germans including Nazis immediately after WWII.

For very brief overview of Argentine history: Citation: "Central & South America" by Kopka, Deborah (2011). Lorenz Educational Press. ISBN 978-1429122511.

For economic and political history: Citation: "Essays on the Economic History of the Argentine Republic" by Díaz Alejandro, Carlos F. (1970). Yale University Press. ISBN 978-0300011937.

Did Trotsky ever stay in Argentina? I've never heard of this. He lived in exile in Mexico in his final years but I've never heard of him living in Argentina.

There was, and still is (8% of total population) a significant German emigrée community in Argentina. I believe I remember reading that there are still German-only language newspapers there. The US was not the only country to receive a large influx of German immigrants. For Eichmann, a substantial community of German emigrées who (I believe) had been at least generally friendly to the German regime would have made that area a prime one to hide out in. And, of course, one that is very far away from his previous location.

For Trotsky, the majority of those Germans in Argentina are descended from "Volga Germans," who were German speakers that came from, well, the Volga River region of Russia. Perhaps this is why Trotsky might have been able to hide there.

Follow up question did any ex-Nazis play a role in the Falklands War?

Related: what was the Argentinian government's response to the Mossad capturing ex Nazi leaders in Argentinian territory?

Alongside everything already mentioned, it's also worth pointing out that Peron saw great value in German scientists and took the opportunity to woo fleeing Nazis to Argentina to attempt to create sustainable nuclear power for Argentina. The project failed but not before a lot of scientists emigrated and settled in Argentina, mostly safely made anonymous by the already established German expat community.

For more information about the Nazis who fled to South America, Hunting Evil is a very good book that explains how and why Nazis fled to Argentina and Brazil to start over.