Boddle, Bubbles's brother, is the curator and owner, and had it built specially to honor the Yoshis. The Yoshi Theater is also the only place where Yoshis can be seen in the game.
The Yoshi Theater first appears during the quest to retrieve the four Beanstar pieces; one of them ends up in the possession of Boddle, who is overseeing preparations for the theater along with his assistant. The Mario Bros. head to the theater and find Boddle using it to decorate the sign. Despite his assistant's best attempts to stop them, Boddle listens to Mario and Luigi's request for the Beanstar piece, and agrees to trade it if they find seven Neon Eggs to decorate the sign with. The Bros. are then directed to Fava, who is found within the theater and provides them with his expertise on Neon Eggs, tasking them to find Bean Fruits for the Yoshis.
Once the seven Neon Eggs are given to Boddle, he places them on the sign and has it hung up as the Yoshis celebrate. He and his assistant then retreat within the theater to prepare for the grand opening. At the end of the game, it is revealed that the events of the plot were made into a movie that is shown in the Yoshi Theater, and the Yoshis cheer wildly at the ending.
In Mario & Luigi: Superstar Saga + Bowser's Minions, the Yoshis watching the events of the game as a movie in the Yoshi Theater is now established in a pre-intro cutscene. The Yoshis also watch the events of Minion Quest: The Search for Bowser unfold within the theater.
All of the movies showing at the Yoshi Theater are based on games within the Nintendo universe (including Mario & Luigi itself).
Legend of Stafy – Based on the The Legendary Starfy video game series.
Wario – A Wario-based movie whose poster is based on a piece of artwork from the WarioWare series.
M & L – It is implied that this is a movie of the very game that Yoshi Theater is featured in: Mario & Luigi: Superstar Saga.
Kirby Story – A seemingly generic movie featuring Kirby. It could be a pun on Mario Story, an alternate name for Paper Mario, as that game's main plot device is the Star Rod, which initially comes from the Kirby series.
* Exclusive to Mario & Luigi: Superstar Saga. • ** Exclusive to Mario & Luigi: Superstar Saga + Bowser's Minions. † Enemy, boss, or stat exclusive to the main game. • †† Enemy, boss, or stat exclusive to Minion Quest: The Search for Bowser. • ††† Enemy variants differ in Mario & Luigi: Superstar Saga + Bowser's Minions.