The issuing of a ban or Silence is automatic based on how many people reported you.
If you go to your e-mail that you got detailing your ban, you can use the link there to get to the support portion of the site and submit a web ticket. If you ask nicely, Blizzard will have a GM review your complaint as well as give you a quote about what you said that cause the trouble.
I’ve been banned several times and each time Blizzard’s GM was nice enough to explain to me why, yet kindly but firmly refused appeals (Which to my understanding, is standard procedure as getting an overturn is very, very rare.).
If you really want to try and get a banning over turned, you can call Blizzard’s customer service hotline and the phone reps will often be more willing to issue an over turn as they are actual customer service reps as opposed to GM staff (Their goal isnt to keep order so much as to do good company PR.).
However the best policy is to simply take your medicine and try to behave better next time. This is a strict forum, and it can be difficult to adjust, I am totally sympathetic to that believe me lol, but it can be done.