Much of Aristotle's Physics is concerned with what nature is, and the reasons behind the way it behaves. He defines four "causes" -- formal, material, efficient, and final -- that answer the question of why an object exists in the form that it does. Though the first three are important in their own right, special priority is given to final cause: that which answers the question "on account of what?", explaining why something exists in the first place, in this context specifically in the natural world (194b, 20). This is the argument that Aristotle presents for final cause in nature; that all aspects of a naturally-occurring object or creature have a particular purpose that led to their being formed the way that they were.
Before its properties and its causes might be addressed, nature itself must be defined. Aristotle does this early on, dividing "nature" into two constituent parts: "the primary underlying matter in each case, of things which have in themselves and source of their movements and changes", and "the shape and form which accords with a thing's account" (193a, 28-31). That is, to consider nature as a whole one must take into account both matter and form. But, he goes on, "the form has a better claim than the matter to be called nature", because it is the form that identifies any particular thing as one of its kind rather than merely having the potential to become it (193b, 8-9).
It follows by logic that if form takes precedence over matter, then material cause is of lesser importance than the other three, because they can all be linked to form itself; formal cause is the shape that something takes, efficient cause is what makes it that shape, and final cause is the purpose for which it exists, whereas "matter is something relative to something, for the matter varies with the form" (194b, 10-11).
Aristotle also notes that "what a thing is, and what it is for, are one and the same", and that "that from which the change originates is the same in form as these" (198a 25-27). That is, formal and material cause can be considered together, as both pertain to purpose; also, in the natural world where matter contains within itself the potential and impetus for change, efficient cause can also be grouped with the others. This leaves final cause standing alone, to be considered separately.
The bulk of the argument for final causes in nature begins when Aristotle, as is his wont, puts forward its opposite: "why should we suppose that nature acts for something and because it is better?" (198b, 17). He paints an image of a natural phenomenon, the rain, which falls not specifically to help the corn grow in the fields nor to make harvested corn in storage rot, but instead "of necessity"; the fact that the rain allows the corn to grow (or to rot, for that matter) is "merely concurrent" with its falling (198b, 17-20). Another example Aristotle gives in the same vein is that of teeth; is it possible that the front teeth could have become "sharp and suitable for biting, and the back teeth broad and serviceable for chewing the food [...] by coincidence?" (198b, 25-27). The question implies that final cause might not be necessary, or even extant, if the same end result might be achieved through blind chance.
To refute this proposition, Aristotle alludes to Empedocles' "man-headed calves" (198b, 32); according to Empedocles, these creatures were formed when the motive force of Love brought together various pieces of flesh and united them in whatever way they happened to collide with each other. As they were not formed "as if they had come to be for something", they did not survive to procreate (or to be observed by Aristotle and his contemporaries). In this way the argument against final cause in nature is established, only to be torn down immediately afterward.
Though Aristotle concedes that this existence-through-chance "might give us pause", his refutation states flatly that "it is impossible [...] that this should be how things are" (198b, 33-35). The first logical reason for the refutation is that "all things which are due to nature [...] come to be as they do always or for the most part", meaning that if the creation of fleshly things was truly as random as Empedocles thought, the relative similarity between individuals of a given kind (or species) would be impossible. Also, the number of aberrations (like the man-headed calves) is remarkably low, to the point of being negligible; this cannot possibly be coincidence, nor can it be automatic (198b, 35 - 199a, 1). Since the other possibilities are exhausted, the fact remains that these natural things (animals, or plants) must have a purpose; "they must be for something" (199a, 6).
Consideration of creatures themselves is followed by considering their activities. In looking for reasons why spiders can construct their webs, or why a plant produces leaves, Aristotle notes that as these are incapable of reason themselves they must be guided by some sort of internal intelligence -- and not only is it in their nature to build webs or grow leaves, it is also for a specific end (perhaps that of catching smaller insects for food, or of protecting fragile fruit) (199a, 27-31). As was established earlier, nature itself is both matter and form; but if the form is tailored to the end result, and the matter follows after the requirements of the form, then that for which the thing in question exists must be its cause (199a, 31-33). This is Aristotle's final cause.
More proof is offered in relation to the world of plants; they, too, have a final cause, though Aristotle concedes that it is "less articulate" than that of the animal kingdom (199b, 10). By this he means that though the features of plants might be less obviously derived from (and suited to) external necessity, they still came about in the same way, to serve their own final cause. Aristotle strikes down the counterargument that favours chance over cause again, by likening Empedocles' man-headed calves to "olive-headed vinelets"; the plant example sounds even more ludicrous than that of the calves, but by logic if the latter was thought to be true then the former ought to have been as well (199b, 11-13).
Aristotle's argument concludes with emphasis on the teleological aspect of the concept of final cause: that everything that exists in nature is proceeding toward an end, unless there is some impediment. The end result which is being sought is not guided by external principles -- those are reserved for the realm of art, not nature -- but instead by internal intelligence, a guiding principle contained within a creature or object that causes it to become the way that it is. For Aristotle, the ultimate final cause of all nature was Plato's "good", or the divine, seeking to realise forms; there can be no explanation within nature for something that is able to transcend it and move other things without being moved itself.
The quotations above were swiped from a course reader, translated from the Greek by William Charlton, and all of them come from books I and II of the Physics (here as published by Clarendon Press, 1992, with eminently useful line numbers!). A different (and arguably better, or at least it would be if it had numbers in the margins for reference) translation of same can be found at http://classics.mit.edu/Aristotle/physics.html.