The main theme of Plato's Allegory of the Cave in the Republic is that human perception cannot derive true knowledge, and instead, real knowledge can only come via philosophical reasoning.
In Plato’s example, prisoners live their entire lives in a cave, only able to see shadows. To them, these shadows are reality. When a prisoner randomly guesses the next shadow to appear, that prisoner will be worshipped as having mastered nature. When one prisoner escapes and understands that life comes from the sun, he realizes his former view of reality based on human perception was wrong. When he returns to tell the other prisoners, they do not believe him, because they are still relying on their perception.
This idea also begins to uncover other issues with human perception. Humans are biased individuals who will explicitly or implicitly apply individual biases to what they perceive. Additionally, two humans can perceive the same object and derive a different meaning from it. For example, a rainstorm can be viewed as a blessing by a farmer or a curse by a sailor.
Ultimately, Plato’s point is that in order to truly gain an understanding of knowledge, humans must submit to the idea of specific and fair philosophical reasoning, which transcends the pitfalls of human perception.
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The Allegory of the Cave represents an expression of Plato's philosophy of truth and reality (which can be termed as Platonic Idealism). When reading Plato's various dialogues, one will often observe a focus upon themes such as justice, virtue, beauty, and the good. For Plato, these various concepts are not human inventions, but are actually built into reality itself, as part of a more transcendent reality (one that is actually more real, in a Platonic sense, than the material world as it is experienced by human beings).
The Allegory of the Cave serves to illustrate this relationship in more concrete terms, with the prisoners serving as a representation of human existence. Just as the prisoners are ultimately trapped in their cave, watching shadows on the wall, so too are human beings trapped by their material existence (with all of our knowledge and assumptions of the world being much like those shadows). Yet truth still exists. Even as the prisoners remain trapped in the cave, there is still the world outside the cave. This represents the reality of transcendent ideals which Plato speaks about.
Thus, this entire allegory sketches the Platonic vision of reality, in which human beings are only able to indirectly grasp the higher, transcendent reality which emanates through the universe.
The answer above offers a fine description and explanation of Plato's allegory of the cave. To expand on it a bit, the theme of the cave allegory gets to the heart of the divide between science and philosophy in Western culture. Plato says we on earth live in a cave, watching shadows on a wall. What we think is real—the natural world—is a pale and imperfect reflection of an ideal reality. He wants us to...
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understand this so that we realize that philosophy is superior to science.
For example, we may think the table in front of us is a real table, worthy of study, and not merely an imperfect imitation of an ideal table. However, we can use the following thought experiment to show that any table we try to build in this world, no matter how perfect we try to make it, is just a shadow of the ideal table. Say we decide to build our table to 1/16" accuracy. That is crude and imperfect: we could always build it to 1/32" accuracy. But that too is crude and imperfect: why not build to 1/64" accuracy? Or for that matter, why not build to 1/100,000" accuracy? But no matter how accurate our measurements, we could always make the table more accurate: hence it will never, in this world, be the ideal table.
Therefore, according to the allegory of the cave, studying the natural world (science) is less worthy than studying philosophy because a scientist is always studying a shadow, an imperfect imitation of the ideal. The only way to truly contemplate reality is to go inward, to use the human mind to imagine the ideal, which is the province of philosophy.
The main theme of Plato’s allegory of the cave is that we humans tend not to understand the true reality of our world. We think that we understand what we are looking at and sensing in our world, but we really just perceive shadows of the true forms of the things that make up the world.
In the allegory of the cave, prisoners are chained in such a way that they can only see the back wall of their cave. They see shadows on the wall of the cave that are cast by objects being moved between a large fire behind them and the wall in front of them. Because they can only see the wall, they do not know that the objects on the wall are just shadows. They think the objects on the wall are the real things. For example, if they saw the shadow of a man, they would think that it was a real man because they had no way of knowing that it was just a shadow.
Plato is saying that, unless we become educated, we human beings are like the prisoners in the cave. We think that we understand the world around us. We think that the things we see and otherwise perceive are real. However, we are incorrect because the things that we perceive are mere shadows. There are true forms of everything that we think we perceive, but we cannot see those forms. In the allegory of the cave, Plato is trying to make us understand that we see shadows and we think they are the real thing. Thus, the main theme of the allegory is that we are ignorant about the true nature of reality.
Further Reading
What would be a viable interpretation of Plato's allegory of a cave?
In The Republic, Plato actually has Socrates provide an explanation for the allegory, stating:
The visible realm should be likened to the prison dwelling, and the light of the fire inside it to the power of the sun. And if you interpret the upward journey and the study of things above as the upward journey of the soul to the intelligible realm, you'll grasp what I hope to convey.
This entire allegory serves as a metaphor to express Plato's understanding of human beings within the physical world and how this relates to that higher realm of ideals which, Plato believed, permeated downwards through reality. Remember, Plato was an idealist—one who had strong tendencies towards Mysticism (as can be seen, perhaps most clearly, in his discussions of the soul and the immortality of the soul). For Plato, material existence is, in truth, only a small facet of reality. Given that material existence is, as far as Plato was concerned, only a small facet of human existence (due to the soul's own non-material nature), this allows human beings some degree of access to that higher, more transcendent level of truth.
The Allegory of the Cave stands as an extended metaphor expressing this understanding of reality (and of human beings within reality). The vision of the prisoners trapped together in the cave, watching shadows being cast on a wall, represents human beings in their material existence. The prisoner who is released and travels into the sunlight represents someone who, through education and engagement in the philosophical life, has come to attain a more advanced understanding of reality itself and has managed to grasp this more transcendent vision of truth.
What would be a viable interpretation of Plato's allegory of a cave?
Socrates and Plato sought knowledge for its own sake and were dedicated to its pursuit. As philosophers, they were deeply interested in the distinctions between objective and subjective realities; in effect, between scientifically-arrived at conclusions and those inferred from incomplete data. Such is the prism through which one reads Plato's "Allegory of a Cave." In this mock dialogue between Plato's mentor, Socrates and the character of Glaucon, Plato attempts to illustrate the manner in which "reality" is falsely perceived and the effects on the subconscious of sudden enlightenment through exposure to an objective reality. In other words, Plato is explaining through the use of metaphor the struggle for objective knowledge and how distinguishing what is real from what is only perceived to be real can create conflict within and among people.
In "The Allegory of the Cave," Plato uses the darkness and isolation of a cave as a metaphor for the closed mind. The prisoners believe only what they see, which are shadows cast upon a wall by the interaction of a fire behind them and walkway passing through the cave, between the prisoners and the fire. The fire casts shadows of who- or whatever crosses the walkway upon the wall, to which the prisoners are forced to stare. As "Socrates" explains to Glaucon, "To them [the prisoners], I said, the truth would be literally nothing but the shadows of the images."
The conflict in Plato's metaphor emerges when one of the prisoners is set free and dispatched into the bright sunlight. Leaving the darkness of the cave for the brightness outside, the prisoner experiences the physical pain that is normally associated with sudden exposure to the sun following a protracted period in the dark. Freed from the physical -- and, in Plato's metaphor, intellectual -- confines of the cave, the prisoner is exposed to objective reality: life as it is, and not as it appeared in the confined space where all of reality was defined by the shadows cast upon the cave's wall. The effect of this exposure to objective reality, as commonly occurs when individuals are disavowed of incorrect notions following exposure to scientific fact, is initially resistance to information that contradicts earlier preconceived notions of reality. Again, Plato explains this phenomenon as such:
"At first, when any of them is liberated and compelled to stand up and turn his neck around and walk and look towards the light, he will suffer sharp pains; the glare will distress him, and he will be unable to see the realities of which in his former state he had seen the shadows; and then conceive some one saying to him, that what he saw before was an illusion, but that now, when his approaching nearer to being and his eye is turned towards more real existence, he has a clearer vision . . ."
The prisoner's subsequent conflict with his fellow prisoners who have remained chained in the cave and whose "reality" remains confined to their interpretations of the shadows on the wall provides the allegory's most important point: that people remain resistant to objective facts when all they have known are subjective interpretations of reality. As Plato/Socrates suggests with respect to the prisoner's enlightenment upon release from the darkness of the cave:
"Last of he will be able to see the sun, and not mere reflections of him in the water, but he will see him in his own proper place, and not in another; and he will contemplate him as he is."
What is the conventional interpretation of Allegory of the Cave in Plato's Republic?
In Plato's Republic, according to the conventional interpretation, the allegory of the cave is an extended metaphor used by the Socratic character to explicate what is usually called "the middle theory of forms", a theory about the nature of reality and knowledge held by Plato in his middle period, but not fully formed in his early dialogues and rejected in his later dialogues. It is extremely unlikely that such a theory was held by the historical Socrates, even though Plato uses Socrates as a mouthpiece to articulate it.
The 'forms" or "ideas" are non-corporeal ideal realities which the objects of everyday life imitate. Just as the people in the cave only see vague monochromatic two dimensional shadows and understand little of the actual world outside, so our ordinary experience is only that of vague and misleading imitations, or shadows, as it were, of the transcendent forms.