Katamari Damacy-Kun and its sequel Oi Katamari Damacy-Kun were released through NTT DoCoMo's i-Mode service (and other Japanese feature phone providers like Yahoo! and EZWeb) within the mid-to-late 2000s, and were inspired by the 2D mini-game from the ending of the PSP game Me & My Katamari. As is the case in that game, players will take control of the prince from the main series and travel across several auto-scroll stages, attempting to roll up as much stuff as they can to achieve a new high score.
These games were only ever released in Japan or were never ported to other platforms, meaning the only way to play them in the past was to try and track down a feature phone from someone in Japan who had downloaded them back in the day. However, now a group of feature phone preservationists have somehow managed to acquire and preserve the files for the i-Mode versions of the games online, making them playable again for a whole new generation via emulation.
To get them to run, you'll need to download the DoJa 5.1 SDK emulator (you can find basic instructions on how to set up the emulator and a download link on this helpful Google Doc). As RockmanCosmo notes in their thread, the process of getting Oi Katamari Damacy-Kun to work is a little more involved than the other i-mode games that have been preserved in the past (on account of it having a specific launcher app and three "stage application" packs), but it essentially requires you to set up the Oi Katamari folders in the SDK's "apps" folder in a specific way before running the launcher.