What I find so fascinating about this essay is how relevant it is in today's world, in which we have 1% holding a disproportionate amount of wealth and 99% holding what is left.  Carnegie's argument is that this is a good thing, as he says,

It is well, nay, essential for the progress of the race,that the houses of some should be homes for all that is highest and best in literature and the arts, and for all the refinements of civilization, rather than that none should be so. Much better this great irregularity than universal squalor (1).

He has set up a dichotomy in which there are two choices, to have "squalor" for all or wealth for a few people. 

As the above response notes, this is meant to afford the masses some "civilized" help or pleasures, to be administered in a paternalistic manner.  This is very much akin, in my opinion, to the trickle down theory of today.  If we allow the wealthy to do as they see fit, 

The price which society pays for the law of competition, like the price it pays for cheap comforts and luxuries, is also great; but the advantage of this law are also greater still, for it is to this law that we owe our wonderful material development, which brings improved conditions in its train (4).

A great deal of the essay is devoted to a justification for wealth, and there is a strong whiff of Puritanism, the idea that the wealthy are worthy of God's favor, as evidenced by their wealth, while the poor, since they have nothing, are not favored by God. As Carnegie discusses, for example, how to help the poor, he says,    

Those worthy of assistance, except in rare cases, seldom require assistance. The really valuable men of the race never do, except in cases of accident or sudden change (17).

People who are "worthy" and "valuable," i.e., graced by God, do not need the help in the first place. 

I do hate to bite the hand that feeds me, since I use my Carnegie Library all the time, but the arguments and tone of the essay are off-putting to me in exactly the same way today's trickle-down theory is. The more things change, the more they remain the same.  

The central theme of Carnegie's argument about the Gospel of Wealth is that rich people are superior to all others and should be allowed to use their money to help the less "fit" people.

Carnegie believed in Social Darwinism.  He believed that the fact that a person was rich showed that he was more fit than others.  This meant that the rich man was the one who knew the most about how to prosper in society.  Carnegie believed that rich people should use their money to help the poor.  But he did not think that they should just give the money away because the poor would not use it in the best ways.  Therefore, the rich man should be paternalistic.  He should spend his money for the benefit of the poor, just as parents spend for the benefit of their children without actually letting the children have the money.

This article, then, is a mixture of Social Darwinism and philanthropy.  It is about how the rich should use their money and their superiority to help the poor.