The expression is referring to an implicit or unstated rule. Defining it is hard but people know when it has been broken. Hooters is an exception, the rule is, don't be like Hooters.
https://en.wiktionary.org/wiki/exception_that_proves_the_rul...
Just as you say, the point is that a rule is implied by a specific exception, as in the example "free entry on Sundays", which implies the unstated rule "pay for entry on other days".
The exception weakens the rule, it's true, but may also reveal the rule.