Note that the philosopher in question is someone who has knowledge (not, e.g., Socrates!). So we get the picture of what that philosopher looks like by an account of what knowledge is.
Knowledge and belief are powers; powers distinguished by their objects (e.g. sight, by visible objects; hearing, by audible objects); so want to know what the objects of knowledge and belief are. Answers: the objects of knowledge are what is; the objects of belief are what is and what is not; (objects of ignorance are only what is not). Note Parmenidean tenor of this.
How to understand this? Some Platonists say that the objects of knowledge are a different realm of reality--the Forms--which are, and are different from the physical objects that both are and are not.
Why or how is a physical thing not? Some Platonists answer, along Heraclitean lines, that: physical objects always changing, so both are and are not. Alternative answer: knowledge always true (always is), belief can be true or false (is or is not), ignorance always false (is not).
So want the philosopher, since he grasps the nature of justice, to rule the city, if we want the city to be just. (Analogy by navigator of ship.)
New idea: philosopher also has knowledge of the Form of the Good, by which he knows what is good about the other forms. Form of the Good like the sun. Sun makes other objects visible by its light; so the Form of the Good makes other forms intelligible to soul and gives them being. The Form of the Good as "beyond being, dignity, power"--a super-Form.
Glaucon objects: what is this Form?
Socrates' answer: the divided line. Shows that the Good is a first principle for Forms: justice, beauty could not be what they are without the Good. So Good is at the top of the hierarchy of Forms, then the Forms themselves, then (below the divided line) are physical objects that partake in the Forms (i.e., are "images" of the Forms), then below them are images of physical things--shadows, etc--(hence, images of images).
So above the divide is what we can know; below is what we can see. Question of what exactly fits where above the divide: just above the line are "hypotheses", but at the top appears to be the Form of the Good. Hypotheses not clearly the Forms, though, so maybe the Forms are in the fundamental principles part (the highest segment).
Cave allegory another illustration of the theory.
Note cave allegory does not commit Plato to a separate "heaven" of Forms, radically separate from physical objects. But they're not radically separate: can get from one to another, and the images are images of real things. So there is a connection. So Plato need not be saying that physical objects are pseudo-real. Plato can be read such that they are just as real.
The alternative reading: there are two ways of looking at the world. We can look at it as a collection of objects of belief (when we look at physical objects, etc) or we can look it as a collection of objects of knowledge (when we think about the Forms). So there's an epistemological, not a metaphysical, separation between the physical and the Forms.
Note also: the cave allegory is not mysticism (as some Platonists think): the prisoners don't get a mystical flash of insight; what the allegory of coming into the sunlight is an allegory of is education (albeit difficult education).