Why did the ancient people who would become Native Americans come to the Americas?
This question is still debated among archaeologists and anthropologists.
Recent findings have overturned the old theory that there was only one migration of the Clovis people which resulted in the colonization of all of the Americas; we now know there were at least three distinct major migrations that occurred.
These migrations began around 15,000 years ago, and continued until at least 14,000 years ago, possibly longer.
It's important to recognize that even though we speak of "a migration", these were probably not organized expeditions with the goal of colonizing new lands. (That sort of thing would not happen until civilizations became organized, as "recently" as 4,000 years ago.) More likely, there were many individual tribes who lived near each other, sometimes in harmony, sometimes in conflict, and as each tribe grew they would sometimes find that they no longer had enough room for everyone.
People would then leave their ancestral lands and try to find new homes elsewhere. Over hundreds of years, a series of these small migrations adds up to a very large migration. We think the reason it happened around 15,000 years ago is that during this time, the Bering Strait was free of ice and the water level was low enough that people could basically walk or swim across.
Another motivation was likely food. In the absence of previous human habitation, wildlife in the Americas was extremely plentiful. There is a long pattern in human evolution of migrants arriving in an area, and gradually driving all the large mammals to extinction---usually by eating them. Human-caused extinction is nothing new; we've been doing it since at least 40,000 years ago in Africa and Asia. Many migrants may have come to the Americas because they found more plentiful animals to hunt there.
In fact, it is also possible that some of the migrants really were trying to explore new lands, simply out of curiosity for what might lay beyond. This is, after all, something humans like to do. We travel to new continents, we climb mountains, and now we even land on the Moon and send robots to other planets. Some of these ancient explorers may have traveled thousands of miles before setting down roots permanently.