This section is one of the most important in the Phaedo, and, frustratingly, it is very confusing. This confusion is not helped any by the fact that Echecrates and Socrates' interlocutors all agree with Socrates unreservedly. We noted this problem earlier--that in his middle dialogues, where Plato first introduces the Theory of Forms, he presents it as totally unobjectionable. (It is only in his later dialogues, such as the Parmenides, that the theory is subjected to scrutiny.) Here, Plato presents it without any kind of detailed or definitive background, assuming that we all understand and agree with what he's talking about. We are not given an explanation of what a Form is (and, we should note, Plato only once uses the word "Form"), we are just asked to agree that there is such a thing as, say, Beauty in itself.
Forms here are being used as causal explanations, something that might cause some confusion. Today, we think of a cause as acting upon something, being that which produces certain effects. Our paradigmatic example of cause-and-effect is one billiard ball causing another other to move. In ancient Greece, there were several different notions of causation, and Plato is mostly concerned with teleological causes. These causes do not explain the "how" so much as the "why" and the "what for." The teleological cause of the second billiard ball moving would not be that it was struck by the first ball, but that it is intended to fall into the corner pocket. Plato's interest in teleological causes explains his disinterest in the provision of addition as the reason that one and one make two. Addition may answer how one and one can combine to make two, but Plato is more interested however in the end goal of adding one and one together. His answer to this is Duality, or Twoness. The end goal of adding one and one is to make two.
Nowadays, we associate teleological causes with explaining people's actions: as Plato explains, the reason Socrates is in prison is best explained by the decisions of the Athenian court and of Socrates himself, not by some material explanation that appeals to his anatomy. Unlike our modern scientific desire to give material explanations for material facts, however, Plato wants to give similarly teleological explanations for the causes of non-human phenomena as well.
Plato explains the causal relationship between Forms and objects in the world by saying that things participate in the Forms. It is not entirely clear what he means by this, but there are three likely possibilities of what Forms are according to this claim. (1) Forms as paradigms: Forms are the perfect instance of whatever they represent. For instance, the Form of Justice is the paradigm of justice, the one, most perfect instance of justice in this world. All other things that are just are just only insofar as they emulate, or are similar to, this Form of Justice. (2) Forms as universals: Forms are that which all instances of the Form have in common. The Form of Justice is that quality which all just people have in common. According to this interpretation, one participates in the Form of Justice by sharing in that quality of justice. (3) Forms as stuffs: Forms are distributed throughout the world. The Form of Justice, under this interpretation, is not some separate thing, but is rather the sum total of all the instances of justice that we might find in the world. There is a little bit of justice in me, there is a little bit of justice in you, and if we were to gather all these little bits of justice together, we would have the Form of Justice. Thus, each one of us participates in the Form of Justice by having a bit of the Form in us, like sharing a small piece of a very big pie.
Mention should also be made of Plato's method of hypothesis. The method is actually quite simple and sensible. The idea, if one is having a dispute with someone, is to find a hypothesis that you both agree upon, and see what follows from it. For instance, Socrates and Simmias disagree about whether or not the soul is an attunement, but they agree about the Theory of Forms. The Theory of Forms implies the Theory of Recollection, which is in conflict with Simmias' argument of attunement. If the hypothesis itself comes into question, the two disputants need merely retreat a little to a new, simpler hypothesis upon which they can both agree. Here, the Theory of Forms is the common ground that everyone shares, and so it can serve as the starting point for a proof of the immortality of the soul.