What is the main theme of Plato's "Allegory of the Cave" in Plato's Republic?
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.
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 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.