Chester Bennington had a troubled back story. He had a difficult childhood filled with abuse. This led to his struggles with mental health issues throughout his life. His experiences of pain and trauma influenced his lyrics which were often very emotional and raw.
No, Howard's Mill is not a true story. It's likely a product of the author's imagination or inspired by various fictional elements. True stories typically have verifiable sources and documentation, which is lacking in this case.
I'm not sure specifically what 'Chester's Day Story' is about as there could be many different stories with that name. It could be about a character named Chester and his adventures during a particular day.