After you are done with a dungeon, you'll automatically save the footage of your progress through the dungeon. From here, you'll have the ability to watch your playthrough of the dungeon, or upload your playthrough for other people to watch.
This is another feature of Meowth's Theater. You'll be able to go online to watch other people's uploaded dungeon footage. You'll have the ability to fast forward and reverse through the floors or to load up a specific floor in case you're looking for a specific moment. Multiple movies are downloaded in bulk from the servers, or you can enter a specific ID to search for a specific video that has been uploaded.
When you search for other videos, you get to see the most popular videos, showcasing how manny times they have been downloaded, their length and what Pokémon have been used in them.