"Survival of the fittest" refers to Charles Darwin's theory of evolution through natural selection in his famous work "On the Origin of Species", published in 1859. By 1889, when Andrew Carnegie published "The Gospel of Wealth", Darwin's theory had become used as justification for a number of other (pseudo-scientific) theories. Key among these was the idea of Social Darwinism, which some used to argue that the "fittest" in society become the most financially successful.
This aspect of Social Darwinism almost certainly underlies Carnegie's philosophy, as he argues that the wealthy have "superior wisdom, experience, and the ability to administer [wealth]", whereas the poor are "foolish, impudent, and unskilled". He argues that the rich must guide the poor, who are "unfit" to guide themselves.
Carnegie embraces competition as the primary means to distinguish the "fit" from the "unfit", via (we can assume) the acquisition of wealth. Those who are "fit" will get rich. Those who are "unfit" will remain poor. While some win and some lose, overall Carnegie felt that the system worked to society's net benefit.
In order to properly evaluate these claims, you should first understand how evolution works, and how the underlying principle of natural selection does -- and more importantly does not -- map onto human social and economic systems. A full discussion of this is off topic, but I have provided links that should help out.