"Know thyself" is interesting, because, simply, if Oedipus really had known himself, he likely would have become a successful king and never suffered the tragic events of the play.
Among other things, Oedipus Rex is a play about identity. Oedipus takes on the identity of king of Thebes without knowing the truth of his past. The trouble really started in the kingdom of Corinth, when Oedipus, as a young man rather than a baby, hears the prophesy that he would kill his father and marry his mother (he assumes this means the king and queen of Corinth).
This is the moment that "Moderation in All Things" comes into play. Had Oedipus been able to control his rage at that moment, he might have been a little more level-headed in his decision making. Instead, he storms out of the castle (and kingdom), kills an old man in his path (his real father) and as we know, ends up in the arms of Jocasta.
Later, as Thebes is facing a plague, Oedipus, without moderation, boasts that he will do whatever it takes to rid the city of the pestilence. His pride at this moment, combined with his lack of knowledge of himself, causes him to rain a curse on his own head. Essentially, these two sayings embody the very values that Oedipus lacks, which become the cause of his downfall.