For something like Turtle WoW's custom races and zones that means shipping modified DBC files (the client-side database tables ChrRaces.dbc, CharBaseInfo.dbc, etc), new models/textures, modified Lua for the character creation UI, and map data for new zones. All packaged in an MPQ that players download alongside the client.
As for DRM Blizzard moved away from MPQ to CASC (their own proprietary archive forma) starting in WoD, which makes this kind of modding significantly harder on modern clients. But the classic-era client binaries have been in the wild for 15+ years, so that ship sailed.