Many people believe that Social Darwinism only 'works' as a postulation if everyone starts from a level playing field. So, for example, it could be argued that if a society (such as the British Empire) has made itself 'civilized' over many generations through enriching itself by exploiting the weak and pillaging the natural resources of their native countries, whilst paying nil wages (slavery) or a pittance, then the various countries did not start on a level playing field. By the mid-Victorian age many imperialist countries had already established their wealth by stealing it from the poor. 'The survival of the fittest' may have been the rule in the animal kingdom or the natural world, but as humans we hopefully aspire to more noble beliefs such as love and protection for the weak who need us most.
Social Darwinism was the idea that, in society, the strong would dominate and the weak would be dominated. You can see how that would translate into support of imperialism.
The idea was that the US was strong -- we had the best (it was thought) society, form of government, etc. We were also strong militarily. Therefore, it was our right to take control of weaker countries.
When it comes to imperialism, this idea was sort of mixed with the idea of the "white man's burden." This idea said that because we were superior, it was our job to help civilize the inferior (nonwhite) people.
In these ways, the idea that the US was stronger than other countries helped to justify imperialism.