As mentioned in an above post, 1920 was immediately after World War I, much of the nation was becoming more interested in doing what was "best" for the United States and Americans. I would say that many of the same reasons many are reluctant to allow immigrants into our country now would hold true for the past also.
It's a little harder for us to understand in the modern day since, even though there are a large number of immigrants who come to the United States every year, it's still nothing compared to the tens of millions who moved here between 1880 and 1920, especially when the total population of the country was so much smaller in those days.
So look at the backlash against immigrants today, the resistance to immigration reform, and the growth of citizen border militias, and imagine how much stronger it must have been in the 1920s, when fully one third of the population was either foreign-born, or who had one foreign-born parent.
Most of the immigration spoken of in the previous response occurred prior to the 1920's. The main problem in the 1920's was the devastation of World War One, which so disillusioned so many Americans that they began to look inward, and withdraw again from the world at large. Part of this withdrawal involved a rebirth of nativism, often disguised as "Americanism." This was also the time of Prohibition and Fundamentalist Religion; a sort of "back to the basics" reaction. Many radical activities, bombings, etc. were by foreign nationals. It was during this time that Sacco and Vanzetti were executed for murder; when the primary evidence against them was their foreign birth. This was also the time of the doctrine of Eugenics, and The Passing of a Great Raceby Madison Grant. This era also saw the resurrection of the Ku Klux Klan, which dedicated itself to 100 per cent Americanism.
Americans supported more restrictive immigration laws in the 1920s for many of the same reasons that many Americans support more restrictive immigration policies now -- they were worried about what they saw as a flood of immigrants who were going to fundamentally change American society.
The immigrants who came over in the second great wave of immigration (late 1800s to WWI) were "different." They were mostly from Eastern and Southern Europe. People from those areas were seen by many people in those days as somehow less "white" than Northern and Western Europeans. Many or most of the new immigrants were of the "wrong" religion -- they were Catholic or Jewish. Many were also seen as politically suspect.
For all of those reasons, many Americans feared them. They felt that the new immigrants were not going to become American. They felt that the new immigrants would, instead, change America and force it to become more like them. That is why they supported laws that restricted immigration.