When you are done with your game, you should make a new project on itch.io, then attach the built game. You should be able to run the game without needing to download any additional software (for Unity, this means making a build of it instead of attaching the Unity project file). It is ideal that you get a file that can be run on the website, because people can play it and review it without needing to download any game file.
If you don't finish your game, that's okay! I think most people don't finish the game on their first jam. (Keep it small and give yourself lots of extra time at the end to fix bugs, tweak things, and uploading usually takes a long time)
You don't need proof that you made it, but if you want to record the process, you can. (Just be honest about creating your game)
