One of the very fast explanation would be made by canonical growth model of Robert Solow.
In this model, the countries with different levels of saving converge to their own steady state (convergence clubs), which means that countries with different savings rate would have different steady states. (so the capital level $K$ and income level $Y$ would be different.) In this case, with capital accumulation, there would be an income inequality. This can be a basic theoretical explanation.
Also, there exists some growth models with technological progress. For example, the South-North economic model. Basically, north is supposed to create the technology by investing in high-tech and there exists some spillovers from North countries to South countries. (You can easily find the formal mathematical model in literature.)
By this simple model, there exists a divergence of economies in terms of their steady state levels of capital and income.
Also, there exists another recent findings of Thomas Piketty which states the famous equality $r>g$ which says that interest rates have been higher than the growth rates of economies, which basically states that people having money have earned more money.
Edit : All points mentionned above is about the inequality between countries but at some point, the inequality between countries could also explain the inequality within country. As between countries, inequality rises, probably, inequality rises in a one country (US or other).
Because the inequality in my explanation stems from capital accumulation. So, it is straightforward to say that, people in a given country, having more savings make investment, accumulate capital and this could probably increase income inequality between capital holders (who have savings) and workers (who have not sufficient savings for investment in technology, machines etc.)
Also, there will be some cycles. People having interest rate from their capital will invest these interests in other investments again from which they could increase more their wealth (I mean capital from wealth.) This also will create more income inequality within a country and I think this explanation is valid not only for US but also for other countries.