How The Cache Works
The cache is a tool the Xbox uses to reduce load times when playing games. Whenever the game attempts to load a file, the Xbox will check if it's in the cache. If it's there, the Xbox will load it; if not, it will copy that file to the cache for future use. Loading from the cache instead of copying from the disk takes less time, and results in faster load times. The Xbox stores a game's cache in either the X, Y, or Z drives, which means it has the capacity to simultaneously store caches for 3 different games. If the player starts playing a 4th different game, the oldest cache will get overwritten.
The biggest reason why this is relevant to JSRF is Sewer Skip. Sewer Skip only works on the English Combo Disk version of the game - specifically, the oversight that allows Sewer Skip to work is found in the Mission and Stage files of the Sewers that are on that specific version. If you use those files on another version of the game, then that version will be able to do Sewer Skip, since it will technically be loading the Combo Disk version of the Sewers.
This is where the cache comes in. If you load the Combo Disk version of the Sewers with an empty cache, the Xbox will save the files for the Combo Disk Sewers into the cache. If you then switch to another version and enter the Sewers, the Xbox will see the files for the Sewers already in the cache, and load those instead of the ones on the disk. This works because all files share the same filename, regardless of version.
On a vanilla Xbox, you can only clear your cache by cycling through 3 different games and then relaunching JSRF. On a softmodded Xbox, you can easily edit what's in your cache with your modded dashboard's File Explorer. This makes it possible for those who don't own the Combo Disk to perform Sewer Skip, as they can FTP the proper files to their cache, and run whatever version of the game they have. The proper Sewer Skip cache can be found on the Mods page, or on the Resources section of the JSRF speedrun.com page.
Apart from Sewer Skip, the cache is also relevant when modifying textures for graffiti or characters. When modifying these files, you mustn't forget to delete the corresponding file in your cache, or the game will simply load the old file and you will not see your imported texture. Most tools, such as neodos's Mod Tool, clear the cache automatically whenever they FTP a file. If you're FTPing it yourself, you must manage your cache manually.