In getting a handle on this complex dialogue, it is helpful to keep in mind two things:
(1) The explicit purpose of the discussion is to show the origin and nature of the universe and of human beings (27a), and these two topics are linked: we are less perfect images of the same intelligible Living Thing of which the universe itself is a more perfect image.
(2) Timaeus's discourse has three major sections. The first section, starting at 27c or 29d, depending on whether you count what Socrates calls the "overture" or "prelude" (29d) has to do with Mind or Intellect as cause; the second section, starting at 47e, has to do with Necessity or the Straying Cause; and the third section, starting at 69a, "weaves together" the two to finish the account of human nature.
We begin with a key distinction: some things are the object of knowledge and some things are the object of opinion. What pertains to knowledge is what is and it is always and unchanging; we get knowledge of it by reason. What pertains to opinion or belief is what is generated, and it is always changing and never purely is; opinion or belief arises about it through sensation. Everything generated has a cause that generates it. When a craftsman/artificer (demiourgos) makes something, if he considers what is ever the same, and, taking this as the model or paradigm, makes what he is making to fit this form, this is what we mean when we say something made is beautiful.
When we take the whole beautiful cosmos and ask whether it is generated, the answer is pretty clear: we sense it, and thus it is an object of opinion. It must then have a cause, a something-responsible that makes it, whatever that first originator might be. This artificer must have made the cosmos on the model of an unchanging paradigm, because the cosmos is beautiful; and thus we can conclude that the world "is a work of craft, modeled after that which is changeless and is grasped by a rational account, that is, by wisdom" (29a).
To understand this, we need to have some kind of answer -- even if we cannot rigorously demonstrate it -- to two questions: Why did the artificer make it, and what model did the artificer use? The answer to why the artificer made it is that the artificer is good, and those who are good want other things to be good. This ends up being very important: the cosmos is set up on the principle of the lower imitating the goodness of the higher to the extent and in the way it can.
As to the model or paradigm, it is the intelligible Animal or Living Thing (zoa) which every other living thing participates. Thus Plato's cosmos is itself a visible living thing that imitates this intelligible living thing, by being alive itself, with a body and a soul, and by being a living thing which other living things participate or partition, and by being in some way a totality of the living and therefore unique. Because it is a self-sufficient living thing, it has no need for the appurtenances we usually associate with animal bodies; it is instead a pure sphere whose vital activity consists entirely of turning on itself in thought and in body, in the closest imitation of its unchanging model that a changing thing can have. Within the universe as well, the higher living things, like the gods who are the fixed stars, imitate that model in the way they can.
Beneath the gods are lesser beings, including men, which imitate the same, but are less pure in their imitation, wandering around in body and in thought, ebbing and flowing in force, buffeted around in sensation and motion, like the cosmos and yet unlike.
Plato's artificer, however, is not a cause creating ex nihilo; he is a shaper of materials. And this means that full understanding of the cosmos (and thus of human beings) requires some idea of the material side. The cosmos is made by Mind (nous) persuading Necessity (ananke) to take a form that imitates the intelligible. This latter is the cause of the errancy or impurity that characterizes the way in which the changing imitates the unchanging.
So in addition to the model and the image or imitation of it, we must also have something that receives the imitation, the "wetnurse" (49a) or "mother" (51a) of all generation. This is obviously very difficult to characterize, because it has no nature of its own, but merely receives the impression of everything else, and indefinite something-or-other capable of being persuaded into becoming any changing thing -- fire, water, earth, air. This receptacle is in a constant state of commotion, constantly shifting as it imitates the unchanging; and the artificer's work is to persuade it to shift in just the right way to be an even better imitation. The way Plato puts it is quite famous. There is a constant commotion of flat triangles in the receptacle; the artificer makes it so that these triangles come together to form volumes, namely, the Platonic solids:
It is worth keeping mind throughout all of this that when we act rationally, we are artificers in our own ways, and we, too, use Mind to persuade Necessity to imitate higher things, drawing out depth from the flatness the world would otherwise have.
It also needs to be kept in mind at this point that, however much fun Timaeus may be having as he speculates about how different features of the world were made by the artificer by mathematical construction in a changing medium, this is all going somewhere quite specific. Timaeus is setting up for Critias's account of human beings in the excellent society, which means that all this needs to contribute to an account of the nobility and corruption of human beings. (In this sense, the philosophical work most similar to Timaeus is Spinoza's Ethics. The Ethics is a treatise on human happiness, human beings at their most excellent. But Spinoza starts pretty much were Plato does, with the All, and from there works out the general principles of the world, which allows him to work out the general principles of human nature, which allows him to get to human happiness.)
Thus the artificer makes the cosmos, the living thing containing living things; the artificer also makes the higher living things, namely, divine things, including the gods and the human soul. The gods make the lower things, including the human body. So the gods make a lesser cosmos, the human body, to be set in order by the divine soul; this lesser cosmos has its own "mortal soul" (69c-d), which works not by intellect but by sensation. The divine soul they put in the head, and the mortal soul they put in the torso. The mortal soul in turn has two parts, a higher victory-loving part, which displaces manliness/courage/fortitude (andreia) and spirit/drive (thumos), and a lower craving/desiring part (epithumios). The higher part they put nearer the head, so that when the lower part decides not to obey the commands coming down from the acropolis of reason, the higher part can listen to reason and restrain desire by force if necessary. (Readers of the Republic will note that this is in fact exactly the account of human nature given there, and it is the same structure that is reflected in Socrates' account of the ideal city both in that dialogue and at the beginning of this dialogue.) The lowest parts of the mortal soul do not understand reason at all; they must be directed by images of reason's commands. So the intellect's commands come down from on high, and they are stamped or painted in the gut, and it is these indirect images that the lowest part follows.
From this we can understand what is required for a human being to be whole and beautiful in both body and soul. This requires the proper proportioning of soul and body, so the means of having a good life is clear:
...there is in fact one way to preserve oneself, and that is not to exercise the soul without exercising the body, nor the body without the soul, so that each may be balanced by the other and so be sound. The mathematician, then, or the ardent devotee of any other intellectual discipline should also provide exercise for his body by taking part in gymnastics, while one who takes care to develop his body should in his turn practice the exercises of the soul by applying himself to the arts and to every pursuit of wisdom, if he is to truly deserve the joint epithets of "fine and good." (88b-c)
"Arts" here is whatever falls under the province of the Muses; "pursuit of wisdom" is philosophy; and to be called both "fine" (kalos) and "good/noble" (agathos) was one of the very highest possible compliments in Greek culture. We should therefore imitate the structure of the cosmos as a whole, always exercising like the every-shifting receptacle, but doing so in a moderate and constructive way according to reason, and in particular we should strive to be (in our own way and to the extent possible) self-sufficient, moving ourselves to health and beauty, just like the cosmos itself constantly moves itself in vital beauty.
But more than this, we need to keep in mind that we have an immortal and a mortal part, and act accordingly. Our divine soul needs to be kept our guiding spirit (daimona), because it is akin to the heavens themselves. (The image in which Plato expresses this, that we stand erect because our divine part tends toward the heavens, in which it is, so to speak, rooted, will later be put in vivid form by Ovid in the Metamorphoses, and thence will become a major image in Boethius's Consolation of Philosophy. This sort of philosophical chain-reaction of images through history is something that one constantly finds if one looks at the reception-history of Timaeus.)
If we live a life that follows only the mortal part, becoming absorbed in our cravings and ambitions, we make ourselves as merely mortal and beast-like as possible; if, on the other hand, we devote ourselves to love of learning (philomathia) and true wisdom/prudence (phronesis), keeping our guiding spirit (daimona) in order, we make ourselves as immortal and divine as possible, and become supremely happy (eudaimona). And when we are well-ordered in this way, we have an affinity to that living god that is the cosmos itself; and this is the most excellent life possible.
Timaeus ends by discussing beasts. This dialogue has a view of reincarnation very similar to that of Phaedrus, so this is not a digression, but one way of looking at the opposite side of the coin. Those who study astronomy -- recall that the stars are gods in Timaeus's account -- but go about foolishly assuming that the only things that are known about them are those thing that can be reached by sensation rather than intellectual understanding, tend to become birds. (The idea seems to be that such astronomers wing about in the sky, but cannot reach the heavens themselves.) Men who have no philosophy at all, and do not study the cosmos, become beasts of the earth. (Guided not by reason but by the thumos in their chests, they are bent over, their heads looking at the ground.) Those with even less reason and understanding become serpents and worms, close to the earth. The stupidest and wickedest men become beasts of the waters, fish and shellfish, no longer even breathing pure air. (These last two are the Men without Chests that C. S. Lewis talks about in The Abolition of Man; Lewis, of course, got the image from Plato.)
And so we are prepared to hear Critias's story, knowing now enough about human beings to know what makes them noble and what makes them degraded.
Quotations from Donald Zeyl's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 1224-1291.