The skunk is known for its reeking odour. This is actually a protective mechanism when the skunk is frightened or under attack. It defends itself by covering its enemies in this stench and hence if one ever encountered one, they would most readily keep the tale to themselves. In other words, if one was to anger, frighten or awaken the skunk they would be left to deal with it on their own. With this observation or elders formulated the lesson that when someone initiates something particularly a misdeed, they must take responsibility for the actions. Hence one should not venture into actions that will lead to everyone forsaking them when they have to face the consequences.
The proverb is used to advise people to take responsibility and face the consequences for their actions particularly if they have initiated a misdemeanour. It is a strong rebuke. It can also be used to absolve oneself if there is another that is responsible for having initiating the misdemeanor.