Oedipus's reasons for seeking out the murder of Laius are as follows:
- Because the king of the land (Laius) was murdered, the land itself has become a wasteland.
- Apollo (a god) wishes that the murder be avenged.
- Oedipus reasons that whoever murdered the former king may well murder him, as well, so it's in his own interest to find the villain and ensure justice is served.
To expand on the first idea, the idea that wounding or murdering a king will destroy his kingdom is a common theme in mythology. We have this in Arthurian legend, for example, when the Fisher King is wounded and his kingdom becomes a wasteland, impotent like he is, until a pure knight comes to help him. The same theme is reflected in T.S. Eliot's post-WWI poem "The Wasteland."