There are a variety of reasons for this. I'm going to focus on Argentina, because it played the largest role in this and accepted a ton of Nazi war criminals, including a lot of famous ones (Mengele, Eichmann, Pavelic). Argentina supported the Axis for most of the war, thanks to close ties to Spain, Italy, and Germany. The Axis in return, particularly Germany, promised valuable trade concessions after the war. President Perón of Argentina was an officer in Italy's army during the 1930s, and many big businessmen in Argentina were supportive of the Axis cause - this led to Argentine agents being in place to assist fugitive Nazis after the war. These agents would help with expenses and false documents.

Also, Germans would bring much of their money with them to Argentina - a lot of this was money or valuables stolen from the Jews. Ante Pavelic and his entourage had a horde of valuables that helped get them out of allied-occupied Europe.

Argentina also saw these men as valuable in the perceived upcoming conflict against communism, and it would be better for them to be there than in the hands of the communists (who Perón despised).

To give you a shorter answer: Argentina had a favorable political climate and a friendly government to ex-Nazis. Eventually many would find their way into other countries such as Brazil and Paraguay.