It is very hard to know for certain why a group of people comes to believe in a given idea. If you ask someone why they believe in something, they are likely to tell you that they believe it because it is true. If you try to ascribe reasons why people believe something, you are likely to reveal as much about your own beliefs as you reveal about the beliefs of the people you are trying to explain. Therefore, we can only speculate about why people believed in Social Darwinism.
It seems easy to explain why people who were successful would find Social Darwinism appealing. Social Darwinism offered a justification for their wealth. It told them that they were rich and successful because they were simply better than everyone else. This meant that they did not need to feel guilty about having possibly exploited others. It meant that they did not have to worry that their society might be unfair. They could just accept the idea that they deserved their riches. It seems very clear that this would be an attractive theory because it would make the rich feel good.
What is harder to explain is why other people believed in Social Darwinism in the late 1800s and early 1900s. In a way, Social Darwinism would appeal to people in the middle class because it would explain why they deserved to be better-off than the poor. But it should not have been completely satisfactory because it implied that the middle class people were simply not as good as the rich. I would speculate that the middle class people who believed in Social Darwinism did so because they believed in the American Dream. They believed that they had what it took to become rich and that they would achieve that goal someday. Social Darwinism gave them the ability to hope that their hard work (and the talent they believed they had) would allow them to succeed. They did not have to simply hope that they would get lucky. This allowed them to feel that they were masters of their own destiny. If they worked hard enough, they would become rich because wealth came to those who worked hard and who deserved it.
Thus, we can speculate the Social Darwinism was appealing to the wealthy because it justified their wealth and that it was appealing to the middle classes because it gave them hope for the future.