Levi-Strauss argued that we understand cultures when we understand their rituals and myths. To do this, we must understand the relationship between terms, not the terms themselves.
As the chart in his essay shows, Levi-Strauss works by breaking a myth down into individual motifs, and then sorting the motifs to facilitate comparing them to one another in order to find patterns of relationship (and therefore significance). He begins with a series of events in the Oedipus myth (p. 214; phrases in square brackets are my additions):
For a quick version of the story, see http://www.lawrence.edu/fast/SPURGINT/60A/oedipus.html and http://www.lawrence.edu/fast/SPURGINT/60A/chart.html
This chronological series Levi-Strauss then rearranges on two axes, preserving approximate chronological order on the vertical axis, and introducing categories of similar motifs on the horizontal axis (NB: I have made a few minor adjustments in Levi-Strauss' chart for the sake of clarity):
Overrating of blood relations Underrating of blood relations Slaying monsters Difficulty walking and standing upright
Cadmus seeks his sister Eurpoa, ravished by Zeus Cadmus kills the dragon The Spartoi kill one another Labdacus (Laius' father) = lame (?) Laius (Oedipus' father) = left-sided (?) Oedipus marries his mother, Jocasta Oedipus kills his father, Laius Oedipus kills the Sphinx Oedipus = swollen-foot (?) Eteocles kills his brother, Polynices Antigone buries her brother, Polynices, despite prohibition
Levi-Strauss analyzes the myth first by looking for recurring motifs, such as the motif of killing. But rather than identify killing per se as a meaningful category, he breaks it down further into the two recurring types of killing found in the story: killing a close relative, and killing a monster. This involves a basic procedure of structuralist analysis, which works by identifying pairs of opposites that are taken as defining an area of significance. In this myth, charcters kill only those who are very much like themselves (blood relations) or those who are entirely different from themselves (monsters). Therefore, part of the myth's meaning presumably has to do with this dichotomy.
Note that this dicotomy is not the only or even the most important one that Levi-Strauss identifies. In fact, he places the category of "killing a close relative" in opposition to a class of acts that involve extreme love for a close relative (Cadmus' pursuit of his sister, Europa; Oedipus' marriage to his mother, Jocasta; Antigone's burial of her brother, Polynices). The opposition Levi-Strauss sees between these categories has nothing to do with killing or with incest per se, but rather with the more abstract motifs that he labels "overvaluing" or "undervaluing blood relations". Similarly, killing monsters, a feat of heroic prowess, stands in opposition to some sort of debility (lameness, vel. sim.). For Levi-Strauss, this opposition involves contradictory positions on the question of whether humans are born from the earth: killing earth-born monsters would be a denial, acknowledgement of debility an admission of this origin.
In sum, then, Levi-Strauss reads the myth as being not at all about killing or incest, but about a contrast between the belief that human beings were originally born from the earth, which is the area of concern marked out by the two right-hand columns (and a major element of the Theban cycle), and the knowledge that they are born now of a sexual union between man and woman, the major concern of the two left-hand columns. The point is not so much that the myth solves a problem as that it outlines a problem or a set of problems and provides, in Levi-Strauss' words, "a kind of logical tool which relates the original problem -- born from one or born from two? -- to the derivative problem: born from different or born from the same? By a correlation of this type, the overrating of blood relations is to the underrating of blood relations as the attempt to escape autochthony is to the impossibility to succeed in it. Although experience contradicts theory, social life validates cosmology by its similarity of structure. Hence cosmology is true" (p. 216).
Overrating of blood relations
Underrating of blood relations
Difficulty walking and standing upright