The Banshee wailing is a reflex they cannot control. Who the banshee wailed for rests in their subconscious mind and to reveal the name it must enter into a trance. Banshees have allergic reactions to iron and to invoke a trance and summon the voice of the ancient mourners they must drink pureed liver, which contains iron.
During this phase, the banshee's eyes turn completely white and the voice turns raspy. They reveal the name of the person who will die and at what time. Afterwards, the banshee reverts back to their old persona and throw up what they drank.
The Banshee (from the Irish bean sí meaning "woman of the síde" or "woman of the fairy mounds") is a feminine spirit in Irish mythology, usually seen as an omen of death and a messenger from the Otherworld. She begins to wail if someone is about to die. 
- 1.09 Fae Day