When a paranormal investigator asks a ghost or spirit to show itself, they are essentially asking it to manifest into a shape that they can perceive with their own eyes. That shape can be translucent shape, a barely visible mist, or a lifelike form often referred to as a full-body apparition.
This lines up well with the Webster’s Dictionary definition for “manifest”, which in verb form, is “to make evident or certain by showing or displaying”.