Unfortunately, there is no way to get unbanned if the ban was not applied in error, which does not sound like it was the case.
"Talking shit" can take a number of forms, from minor offenses to serious and unconscionable toxicity, and all of these can earn different punishments, from the 10-game chat restriction (for first time offenders with relatively minor content) to a permanent ban (for repeated offenders or truly awful displays of unacceptable behavior). If you can post your punishment history (if any) and your chat logs we can help you figure out what your specific problem was.
Ultimately, League holds individuals responsible for their own actions and behavior. If someone is creating a toxic environment by insulting you and you respond by insulting them back, all that has happened is that there are now TWO people contributing to a toxic environment. Both of your behavior is then punishable.
The correct approach is to mute him, ignore him, or, if you must engage, do so in a polite and non-confrontational manner. Otherwise you are furthering the toxicity in that game, and subject to the punishments for doing so.
I'm sorry this isn't the news you wanted to hear.