How does Oedipus persuade witnesses to come forth with information about the murder?
Once it is revealed thatthe slource of pollution is the killer of Laius, Oedipus makes a proclamation designed to get iunformation about the murder. He offers that he will let the killer go into exile if the killer confesses and offers a reward to anyone with information about the killer. He sets this out in the speech at 250 sq.:
... I now proclaim
the following to all of you Cadmeians:
Whoever among you knows the man it was 260
who murdered Laius, son of Labdacus,
I order him to reveal it all to me.
And if the murderer’s afraid, I tell him
to avoid the danger of the major charge
by speaking out against himself. If so,
he will be sent out from this land unhurt—
and undergo no further punishment.
If someone knows the killer is a stranger, 
from some other state, let him not stay mute.
As well as a reward, he’ll earn my thanks. 270
But if he remains quiet, if anyone,
through fear, hides himself or a friend of his
against my orders, here’s what I shall do—
so listen to my words. For I decree
that no one in this land, in which I rule
as your own king, shall give that killer shelter
or talk to him, whoever he may be,
or act in concert with him during prayers,
or sacrifice, or sharing lustral water.* 
Ban him from your homes, every one of you, 280
for he is our pollution, as the Pythian god
has just revealed to me.