An exception is something that does not follow the rule. When we say that the exception proves the rule, we mean that excepting those cases shows that the rule exists, or that it applies to those not excepted. For example, it is a rule of spelling that when the sound is like ee in meet, we use i before e except after c. The only exception to this rule are seize, weird, counterfeit. That they are exceptions prove that the rule exists and that it applies to all other words containing the sound of ee in meet, e.g. grief, niece, piece.
This saying is often used as a proverb in a most illogical way:
‘The only member of the team to play well was young Parfett, who was the exception that proved the rule.’
What does ‘the exception that proved the rule’ here? It means nothing at all.