I use to be a mod on a moderately large subreddit (over 200k users last I checked). Unfortunately, the only thing you can do is appeal to the mods themselves. It will go to modmail, which is basically a new thread between yourself and all of the mods. They can discuss it with you, and if one chooses, lift the ban. Reddit admins hardly ever intervene, only in extremely severe circumstances. Each subreddit is its own board, governed by individuals largely unsaddled by the wider site rules and regulations.
And of course you can make a new account. With how little personalization there is (though they're adding more) almost no one remembers anyone else's names. Only the really fanatical users. The other ten or so mods and I use to keep a shared Google spreadsheet just to remember who the commonly problem users were. We barely ever banned anyone, rather gave warnings and tried to guide discussion in more favorable avenues, because what's the point when anyone can just make a new account? But mods of other subreddits are dicks and ban people for even slightly disagreeing with the narrative. /r/conservative and /r/the_donald were definitely the worst offenders. If you weren't a 100% true believing cheerleader you'd get banned with often obscene or curt and banal messages.
