In his opening speech, the priest compares Thebes' present plague to the troubled times the city was in the midst of when Oedipus first came to Thebes. Years ago, the city was being plagued by a Sphinx. According to Greek mythology, a Sphinx is a creature that has a lion's body, bird's wings, and a woman's face. It speaks in riddles and whomever it approaches that cannot answer the riddle is eaten alive. According to legend, a Sphinx was sent to Thebes by the gods as a plague and Creon promised to make king anyone who could solve the Sphinx's riddle ("Sphinx," theoi.com). Since Oedipus rescued the city from the Sphinx so many years ago, the priest is begging Oedipus to rescue them from their present plague, thereby comparing both plagues.

We see the priest allude to this past plague in his lines:

It was you who came and released Cadmus' town
from the tribute we paid to the cruel songstress,
and these things you did knowing nothing from us,
nor instructed at all, but with help from god
you spoke and knew how to set out lives straight. (39-43)

Hence, since the priest believes that Oedipus was sent by the gods to rescue them from the Sphinx, he believes that with the help of the gods Oedipus can heal the present plague.