At a time when science fiction movies were invariably cheap rubber monsters attacking our cities and scaring our women, "Forbidden Planet" offered an usually thought-provoking plot that worked on a number of levels. Today, too many sci-fi movies are nothing but computer-generated special effects extravaganzas masking the lack of thoughtful plot and characterization. "Forbidden Planet" had awesome special effects for their time (many of which still hold up well today)--but these were used to effectively support the multifaceted plot and characterizations, not try to compensate for their lack.
The Shakespearean ("The Tempest") and Freudian ("Id-monster") elements have been noted by many critics. In the 1950's, with the atomic and hydrogen bombs so new and terrifying, other sci-fi movies had asked whether man had the wisdom to use all the new science for good rather than evil. But they usually dealt with that solely on a surface level, by just having some monster created with the new science that comes out and kills a bunch of people. Only "Forbidden Planet" dared to actually delve into the depths of human psychology to see what our baser instincts are capable of when given full rein. It directly refuted the notion that all that new science and technology was somehow civilizing humanity. The Krell, a far more advanced race than we, are never seen on screen (only their artifacts are shown, leaving you to imagine what they looked like). But their disappearance is a warning that even a far more advanced race like they, couldn't escape the baser instincts and subconscious drives deep within their own brains--so what of man?
One subplot that is less often discussed, but equally well thought out, is the scenes with Altaira and the tiger, an allusion to the myth of the virgin and the unicorn. Until Altaira meets the male crew of the C-57-D, she is virginal and the tiger is a tame beast in her presence. After she has her romantic interludes with Farman and the captain, the tiger attacks her. (Being the 1950's, the dialogue only subtly suggests what has happened.)
For "Star Trek" fans, it's worth seeing "Forbidden Planet" just to list all the parallels between "Forbidden Planet" and "Star Trek: The Original Series". The basic theme (a "United Planets" spaceship explores a strange new world), the characters and characterizations, the weapons, and even the special effects all seem to have unconsciously inspired Gene Roddenberry to create his own vision.