To answer this, just look at the plot of the film. You have good Americans who are replaced by people who look just like them but who are actually aliens. This can be seen as a metaphor for what many Americans feared that communists could do.
In those days, there were fears that anyone could be a communist. McCarthyism fed this fear--if there could be that many communists in the government, who might not be a communist? You would never be able to tell. This is just like what happens in the film. In the film, it is impossible to really tell the pod people from the regular people, just as it was seen as impossible to tell a communist from a "good American." In the film, the pod people are gradually infiltrating and changing people. This was what people thought the communists could do. They worried that the communists would secretly infiltrate and then start turning other people into communists without anyone knowing it.
In this way, the film reflects the concerns that people had--it reflects the fear that the communists could covertly subvert the whole country, turning good Americans into reds.