The Puritans' success can be attributed to two main factors. These factors are the nature of their religious beliefs and the fact that they came to the New World largely as family groups.
The Puritans' religious beliefs caused them to believe very strongly in hard work and improving themselves and their society. They did not believe that they could achieve salvation this way because they believed their fate was predestined. But they did believe that their actions could give clues as to whether they would achieve salvation. Therefore, they tended to want to work hard and get ahead in life. They also believed that it was important to act in the correct ways as a society so that God would reward the society here on Earth. For these reasons, the Puritans were hard workers, a trait which helped them to succeed.
In addition, the Puritans generally came in family groups and even as whole communities. This allowed them to have a very stable society from the beginning. It helped to create a society in which people got along relatively well and had good, happy lives. This is important in their success as well.
We should not forget that the Puritans had a great deal of initial help from the Indians as well.
Overall, though, if we are looking at factors having to do with the Puritans themselves, religious beliefs and the presence of families are two major factors in their success.