Oedipus killed his father, Laius, at a place where three roads meet. A fateful decision made at a crossroads, huh? We detect symbolism. Oedipus could've run into his father anywhere along the Theban Way, but instead Sophocles specifically places the confrontation at a three-way intersection.
Crossroads are a traditional symbol of choice in literature. Makes sense, right? You come to an intersection and you have to decide which way to go. It's probably pretty easy to see how such a place could represent all the moments of choice in our life. In a way, every second in one's life is a tiny little crossroad. Every small choice we make affects our future in someway.
Of course, Oedipus's fate has been predetermined from birth. For their own mysterious reasons the gods have decided that it's necessary for Oedipus to have a tragic life. Oedipus does make a fateful choice at the crossroads, but it is one that he was predestined to make. Perhaps the highly religious Sophocles is trying to show how all the seemingly free choices we make in life are really programmed into us by higher powers.
It's interesting that Sophocles makes it specifically a three-way crossroads. (Why not four or five?) The fact that it's three is reiterated several times in the play, so there's a good chance it has some larger significance. There's not a lot of scholarship on this particular detail, but we have a theory: the three roads represent past, present, and future.
The Greek goddess of the crossroads, Hecate, was said to have three heads. Each head looked down a different path – one saw the past, one the present, and one the future. Though Hecate isn't mentioned in the play, perhaps the three-way crossroads in Oedipus the King has a similar symbolism. This idea is pretty darn consistent with themes of fate in the play. Oedipus is being pushed along by the irreversible flow of time.
Also, note that Oedipus was three days old when his parents abandoned him. Is there some connection between this and the junction of three roads? What do you think?