First of all, they mostly didn't. 75% of the Nazis who went to Argentina ended up leaving, using it only as a waystation, and while Mengele did stay in Argentina for about ten years, he ended up leaving himself, first to Paraguay and then Brazil. (Of course, Eichmann famously remained in Argentina until his capture.)

As to why they started off in Argentina, this answer by u/brassafrax should start you off nicely.