The game is set over two years after Persona 3 FES's The Answer, and two months after Persona 4's True Ending. As a result of Teddie's sudden disappearance in Inaba, his comrades from the Investigation Team begin watching the Midnight Channel, where random people are often seen, in hopes of spotting him. They find what appears to be Teddie in a pirate outfit announcing a fighting tournament called the "P-1 Grand Prix" to see who is the "Ultimate Persona User". Confused by this turn of events, the group agrees to reunite and enter the tournament to find out what has happened to their friend, as well as see who among them is the "Ultimate Persona User". All of the characters are also meant to have their own original storylines.
Source: Wikipedia, "Persona 4 Arena", available under the CC-BY-SA License.