"How did they usurp the South Americans?" The assimilation was very gradual, not dramatic.
"Why did they start building robots? And how?" They needed human-like intermediaries to help them deal with humans. The technology for robots is well advanced by this time. Robots are being built by many developed countries.
"Why did the rest of the world's population let them take over South America?" The takeover was transparent enough that by the time the rest of the world understood what was happening, it was too late. Since they control all of South America (and the weapons of those countries) the other nations fear their strength. There is also a sentiment among some parts of the human populace to leave the dinosaurs alone, "let them have South America, as long as they don't bother us.
The dinosaurs easily grew their numbers; the modifications made to them by the process of bringing them back to life having the unexpected side effect of developing not only their intelligence but also their growth rate. They matured at an increased rate and therefore could reproduce faster.
The process of population expansion occurred over a number of years. As no one visited the island, this went completely unnoticed. As their intelligence developed and they learned to speak, they began to develop a plan to leave the island.
The former occupants of the island left numerous boats, with some of them large enough for larger Triceratops, Brachiosaurus and Tyrannosaurus. The smaller, more agile Dilophosaurus and Velociraptors were obliged to take the smaller craft. To the Gallimimus and Pachycephalosaurus went the task of driving the boats.
The intention, at first, was to simply increase their habitat range, but a small number of them evolved a plan to go beyond just a land-grab and sought to eliminate humans altogether. They kept this a secret from the others.
Plans for the expedition reached a climax in late December, with a group of eager dinosaurs forcing the issue to a vote, which went in their favor. Consensus was for a mass exodus and plans were finalized.
The group decided that Costa Rica, only 120 miles away, was the best place for landfall; specifically the Guanacaste Province, as it was the most westerly point on the mainland. The Cabo Blanco Nature Reserve on the Nicoya Peninsula was deemed the ideal landing point for many reasons, most notably the presence of food and the lack of human populations.
The journey was fraught with disaster, as none of the travelers were accustomed to travelling in boats on water. Sickness was common but none of the party died.
Upon landing on a deserted beach, it was almost impossible to keep the group together. All wanted to explore these new surroundings anf find sources of food. The more dominant Triceratops convinced the majority of the party together. Many, however, did escape to the surrounding cities.
The reports began to surface in March of deaths from large animal attacks but the animals were most unusual: eyewitnesses said they looked like small dinosaurs. And some said they spoke. "Talking dinosaurs" led most network and Internet news reports.
This, of course, was initially ridiculed but once videos of the attackers surfaced, disbelief became anguish. How could they kill and talk? And what was most disturbing was what they were saying: "South America is ours; leave us alone."
Attempts to contact leaders in South America proved fruitless; no one answered the old phone numbers. While leaders of all other countries tried to comfort their people, most of the world's population felt that if South America is "all" they wanted, let them have it. No one, however, was completely sure that the dinosaurs would stop there.
The obvious physical limitations that confront any dinosaur in a world created by man did not stop them. They were able to locate and threaten those who could help them, first forcing them to be their surrogates and then by forcing these surrogates to build them better surrogates that were easier to control: robots.
Robots were common place at the time but society had, after much debate and disagreement, fell in line with Isaac Asimov’s Three Laws of Robotics, briefly stated as 1) a robot cannot harm a human; 2) a robot must obey humans; and 3) a robot must keep itself alive, unless that conflicts with 1 or 2. All of these are superseded by Asimov’s later “zeroth” law, which states that “a robot may not harm humanity.” This would allow a robot to harm some humans to save humanity. These are the robots of the time.
The dinosaur surrogates were required to ignore such guidelines, in favor of fewer restrictions. This was the subject of much debate amongst the dinosaurs, as some wanted to merely subdue humans, while the eventual majority chose harsher tactics. What has soured the dinosaurs towards humans is their poor custodianship of the planet, driving many species to extinction. Having faced extinction already, the dinosaurs do not want it to happen again.
Many humans, seeking to change the pattern of extinction, are willing
to work with the dinosaurs to prevent a final confrontation that could
mean the end of mankind. There were more than enough humans that were determined to see the dinosaurs made extinct a second time.