Darklordoftech and mollybecky both gave good answers. I'll add a few points. First, the Republicans were the sort of naturally dominant party between about 1896 and 1932 (sometimes known as the
Fourth Party System), and 1920 was a reassertion of that dominance after Wilson's fluky victory in 1912 and his narrow re-election in 1916. Additionally, the 1920 presidential election represented what Harding called a "return to normalcy", not only from the Wilson administration but from the entire progressive era. In other words, many people were tired of the activist government of the Roosevelt, Taft, and Wilson administrations.
Also, I think it's important to understand the absolute sh**tshow that was the Wilson administration after the end of WW1. You simultaneously had high unemployment, high inflation, one of the worst pandemics in U.S. history, some of the worst racial unrest in U.S. history, the First Red Scare, labor unrest, a chaotic demobilization both of U.S. soldiers and more broadly of wartime organizations like the Railroad Administration, a polarizing fight over the League of Nations, and, in the midst of all this, the implementation of Prohibition. Any party likely would have been doomed by this confluence of factors, but the situation was made even worse after Wilson suffered a stroke in late 1919 that left him largely incapable of providing leadership.