Ready for some heavy hitting? Louis Menand calls the Cat "a Cold War invention" and an "analyst of the psychology of his time" (source). It's not a tough argument to make: once you think about it, Seuss does seem to be using fears of Communist infiltration as fodder for this tale.
Philip Nel is on the same page. He convincingly argues that Dr. Seuss, who had many Hollywood connections, sympathized with members of the film industry who were being blacklisted and even imprisoned because of alleged Communist affiliations (source, 65).
With that in mind, think of the caged up (and, ahem, bright red) Things and how chaos ensues when they're released. If we can read the Things as stand-ins for the imprisoned movie folk, Seuss might be arguing that it's silly to imagine movie makers as threats to national security. How much trouble can those Things really cause?
But wait a second. Seuss worked on propaganda films when he was in the Army during World War II. That means he knew that film was and is a powerful medium certainly capable of impacting national security.
So what was he driving at? Really, we're asking. History might help you decide.