By suggesting three cards in your own hand, none of the other teams will be able to refute your suggestion. At this point you do not make an accusation and game plays goes on. This event makes it clear to the other teams that you hold either: one, two, or three of the cards from your suggestion.
From the perspective of the other teams, either two, one, or zero of the cards you suggested are part of the solution. Since the other teams do not know you are playing the Exhibitionist Gambit, from their viewpoint it is likely your suggestion holds at least one of the solution cards.
The result is other teams are likely to re-use your cards in their own suggestion (especially the person and/or the weapon, since the room requires they be in the correct spot on the board). If the suggestion is refuted before it makes it back to you, you know the refutation does not involve any of the cards in your own hand, which often provides insight into what card was used to refute.
Increasing the likelihood some of your cards are used in suggestions (and the general confusion caused by an unrefuted suggestion) are the benefits of this gambit.
This gambit can only be played if you are dealt at least one of each type of card (person, weapon, and room) and you can get to the room in question.
My father-in-law holds a virtual Clue game weekly for members of our family. The result of which is our family members have spent a lot of time playing (and thinking) about Clue. At the time of this writing, 38 games have been played. I’ve used the the Exhibitionist Gambit twice (games 36 and 38). In both cases, I won the game.