This proverb is often used to justify some rule you have proposedbut which someone else has listed exceptions.
This are exapmples of use found on the net: Meg: Men are alwaysrude. Caroline: But Fred's always polite. And John and Tony are polite, too. Meg: They're just the exceptions that prove the rule.
Mike: All the shows on TV are aimed at people with low intelligence. Paul: What about that news program you like to watch? Mike: The exception proves the rule.
Anyway, there is a contradiction in this proverb because an exception does not confirm a rule, rather, it proves that the rule is not valid.
The sentence comes from a Latin statement, Exceptio probat regulam in casibus non exceptis, that is to say, when we say that it is prohibited to park here on Sundays, it proves that there is a rule, which is that you can park there on the rest of the days of the week. In this example, the prohibition to park there on Sunday is a exception that proves the existence of the following rule: you can park there.
It is in these contexts where we can use the proverb if we want it to sound meaningful.