That does not make any sense. They just offered him the chance to waive the court process and go back to Ireland immediately. It does not mean that he must be here legally. It just means that a court will have to make that determination before any further steps can be taken.
I know it sounds crazy, but they do need his consent to being deported.
Coming to the U.S. under the Visa Waiver Program means you waive any rights you have to claim asylum, etc. This guy admits he came here under Visa waiver, and that he overstayed his visa by years. But they still can’t deport him, because he’s raising new legal arguments based on being married to a US citizen.