This would be better suited for /r/AskScience, but I can provide you with a basic answer. First, there were placental mammals in South America from the beginning of the Cenozoic. The modern descendants of these first South American placentals are things like anteaters, sloths, and armadillos. Another branch (roughly speaking) of indigenous placentals were a group of ungulate-like creatures distantly related to hoofed animals in the northern hemisphere; these are all extinct today.

Monkeys and South American rodents are the next group of placentals to make it to South America. Genetically, they're closest relatives are in Africa and likely they reached South America by island-hopping / rafting of vegetative mats washed out from Africa via the Congo or other rivers and crossing the Atlantic when it was much narrower.

Perissodactyls (tapirs, horses, etc.) and camelids (camels, llamas, etc.) evolved in North America originally. When the Isthmus of Panama formed, tapirs, horses, llamas, deer, elephants, cats, dogs, etc., went south, while things like armadillos, opossums, and giant ground sloths, monkeys, porcupines, etc., went north.