The Prisoner's Gate is the gate to Castle Heterodyne through which prisoners enter. Like most Heterodyne related objects, it carries the Trilobyte sigil. The doors of the castle are controlled by the Castle's intelligence, as the guard at the door has to call upon the castle to let prisoners enter. The doors have intermeshing spikes that further secure the door, and likely act as a deterrent to would-be escapees.
Agatha entered Castle Heterodyne through this gate to avoid anyone suspecting her of being a Heterodyne. Who, after all, would expect an heir to the Heterodyne legacy of power, wealth and fame to show up in her own castle as a prisoner? Only a true paranoiac would expect such a thing.