A request that the listener/audience waits until they have heard all the information before voicing their opinion. Often said when the speaker anticipates that what they are about to say will be prematurely interrupted, dismissed or ignored without allowing the full story being told.
Family Guy: Brian & Stewie (S08E17)
Stewie: Look, I’m going to propose something and I need you to hear me out. Will you promise me you’ll do that?
Brian: I guess...
Stewie: You guess? Or you will?
Brian: Yes I’ll hear you out.