Socrates is accused of corrupting the youth and teaching gods other than the gods of the state.
In Ancient Greece, a charge of impiety was a significant issue - not just a "religious" offense. The city-state would be in danger of the wrath of the gods. Essentially it was viewed as treason. Very serious charge.
In Ancient Greece, the prosecutor would be the one who had brought the accusation. Therefore, we know that Euthyphro was not just the prosecutor, but the one who brought the charges.
Euthyphro was bring charges of murder against his father. Something (as Socrates points out) would be viewed as an impious action, therefore, Euthyphro must have a remarkable understanding of what piety must be.
"What is piety?"