We may say that the theory of Forms is Plato's solution to the "problem of essence", i.e. Plato's answer to the Socratic question: “What is it?” (Ti esti ... ?)
See Phaedo, 65d-65e :
“Now how about such things as this, Simmias? Do we think there is such a thing as absolute justice, or not?”
“We certainly think there is.”
“And absolute beauty and goodness.”
“Well, did you ever see anything of that kind with your eyes?”
“Certainly not,” said he.
“Or did you ever reach them with any of the bodily senses? I am speaking of all such things, as size, health, strength, and in short the essence [οὐσίας] or underlying quality of everything. Is their true nature contemplated by means of the body? Is it not rather the case that he who prepares himself most carefully to understand the true essence of each thing that he examines would come nearest to the knowledge of it?”
See also the discussion about the immortal soul and the recollection:
[76d] “Then, Simmias,” said he, “is this the state of the case? If, as we are always saying, the beautiful exists, and the good, and every essence of that kind, and if we refer all our sensations to these, which we find existed previously and are now ours, and compare our sensations with these, is it not a necessary inference that just as these abstractions exist, so our souls existed before we were born [...].
[78d] “Is the absolute essence, which we in our dialectic process of question and answer call true being, always the same or is it liable to change? Absolute equality, absolute beauty, any absolute existence, true being—do they ever admit of any change whatsoever? Or does each absolute essence, since it is uniform and exists by itself, remain the same and never in any way admit of any change?”
[79d] “But when the soul inquires alone by itself, it departs into the realm of the pure, the everlasting, the immortal and the changeless, and being akin to these it dwells always with them whenever it is by itself and is not hindered, and it has rest from its wanderings and remains always the same and unchanging with the changeless, since it is in communion therewith. And this state of the soul is called wisdom. Is it not so?”
Finally, see 96a-ff for the discussion of the "real causes" of changing and coming to be, with reference to the "naturalists", like Anaxagoras :
[103e] “The fact is,” said he, “in some such cases, that not only the abstract idea itself [τὸ εἶδος] has a right to the same name through all time, but also something else, which is not the idea, but which always, whenever it exists, has the form of the idea. But perhaps I can make my meaning clearer by some examples. In numbers, the odd must always have the name of odd, must it not?”
“But is this the only thing so called (for this is what I mean to ask), or is there something else, which is not identical with the odd but nevertheless has a right to the name of odd in addition to its own name, because it is of such a nature that it is never separated from the odd? I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called odd, which is not the same as three? Yet the number three and the number five and half of numbers in general are so constituted, that each of them is odd though not identified with the idea of odd. And in the same way two and four and all the other series of numbers are even, each of them, though not identical with evenness. Do you agree, or not?”