Survival of the fittest is part of Charles Darwin's mechanism for evolutionary changes. It's part of a larger concept called "natural selection." It works like this. Random changes in the genetic code lead to individuals in a species having various adaptations. A particular adaptation might be good for a particular time and environment, or it might be bad. Individuals with an adaptation that is not beneficial are more likely to die or be killed. Individuals with a beneficial adaptation are more likely to survive. The ability of an organism to survive and pass on their genes is called "fitness." In other words, organisms with beneficial adaptations are the fittest organisms within a given population; therefore, the fittest tend to survive. Survival of the fittest.
Because the fittest are surviving, they are passing on their genetic traits. Nature is essentially selecting which traits are getting passed down from generation to generation and which traits disappear from the gene pool. Hence, natural selection. I summarize it all for my students with a flow chart that resembles the following:
Adaptation --> increased fitness --> survival via natural selection --> evolutionary changes