From the Super Mario Wiki, the Mario encyclopedia
The paper trampoline is a key item in Mario & Luigi: Paper Jam.
When Mario, Luigi, and Paper Mario chase Nabbit across Doop Doop Dunes, two giant Pokeys fall from the sky, sending the trio flying. When they attempt to cross the bridge leading to Bowser's Castle, the Pokeys slam into it, knocking them into the Doop Doop Dunes Grotto. When they make it back to the surface, they find a Toad Village with a Lakitu Info Center. Starlow asks the Paper Toads if they can make them something to help them jump really high. After completing more Paper Toad Rescue Missions, a Toad arrives with a delivery from Peach's Castle containing the paper trampoline and a new Trio Attack, Trio Kite.
Mario, Luigi, and Paper Mario then return to the grotto, where they put the paper trampoline down and use it to launch themselves onto a high nearby platform, allowing them to fight the Pokey duo. The trio can use the paper trampoline for the rest of the game, and can grab onto the bridge by pressing at the right time.