The title is based on an African fable of the same name about a chicken named Chicken Little who is hit on the head, believes that the heavens are breaking into pieces and runs around shouting "The sky is falling!" The phrase is also a common expression indicating a hysterical or mistaken belief that disaster is imminent.
The trivia item below may give away important plot points.
The top portion of the flying saucer is similar to one from "The Day the Earth Stood Still", (1951). Some of the scenes of frightened people seem to be from that same movie. The craft that shuttles Nelson from the Seaview to the saucer looks like a miniature of the main saucer in the movie "Atomic Submarine", another 50's science fiction movie.