People usually prove it by making the case they haven't done anything terrible within the past couple months.
When someone does eventually get unmuted it's because they've proven that they really haven't done anything to merit concern from the community in awhile.
The best way to prove you're deserving of an unmute is to give people a lack of evidence

. Very rarely will we hold something that you said three months ago against you unless it's consistent behaviour or it's quite alarming specifically.