Primate did originate in Africa around 60 million years ago, right after dinosaur extinction, but soon after, tarsiers, lorises, lemurs and monkeys spread over tropical Africa and Asia.
40 million years ago a few monkeys arrived in South America, maybe on board a vegetation raft. Then they evolved into hundreds of modern New World Monkey species.
25 million years ago, apes speciated among African monkeys. Then they split into great apes (aka hominids) in Africa and lesser apes those migrated to Asia.
16 million years ago, a few hominids migrated from Africa to Asia, then evolved into the orangutan branch
9 million years ago, remaining African apes split into gorilla and hominini.
6 million years ago, hominini split into Homo and Pan (chimpanzees) genera, both in Africa.
1.8 million years ago, humans (Homo ergaster/erectus) spread out of Africa into Asia and Europe, and evolved into various local species/subspecies including H. sapiens in Africa/Middle-East (us), Neanderthals in Europe and some others.
60,000 years ago, a small group of African Homo sapiens spread all over the World.
Monkey/apes evolve everywhere into hundreds of species, 274 are still alive, we are just one of them.
However, in the last 3 million years Africa has faced severe climate changes. Intelligence along with bipedalism and omnivorous diet has been they best ticket to deal with those changes.
The most successful species among all primates is the smartest, the one that can walk long distances and eat whatever they find: animals (hunted with hand made weapons), roots (only digestible when cooked on fire). In cold season, they wear other animal fur (expecting thicker fur from natural evolution is way too long).
Other primates had to stay in their tropical forest, and eventually become extinct with the forest.
Most of them are now endangered species.
@ JZ
You're right, there were quite bipedal hominids 6 million years ago.
However, it seems bipedalism wasn't that beneficial back then. By the way, Orrorin (Orrorin tugenensis - 6 Mya) close to human-chimp split was a better biped than both later Australopithecines and modern chimpanzees. Lucy for example (A. afarensis - 3.2 Mya) walked awkwardly with a rolling gait because of her pelvis-femur geometry, so did very first humans (H. habilis - 2.4 Mya)
Humans became real tall savanna runners 2 to 1.8 Mya (H. ergaster, H. erectus). They also became smarter and started out of Africa migrations in the same period.
The kick-off event seems to be the Isthmus of Panama creation 3 Mya rerouting ocean currents.
This changed dramatically Earth climate: glaciation cycle, creation of ice caps...
Human wasn't the best adapted species to one particular climate, but intelligence, omnivorism and bipedalism gave us a faster adaptability to changing conditions.