"The exception [that] proves the rule" means that the presence of an exception applying to a specific case establishes ("proves") that a general rule exists. For example, a sign that says "parking prohibited on Sundays" (the exception) "proves" that parking is allowed on the other six days of the week (the rule). A more explicit phrasing might be "The exception that proves the existence of the rule."
From http://en.wikipedia.org/wiki/Exception_that_proves_the_rule
"The exception [that] proves the rule" means that the presence of an exception applying to a specific case establishes ("proves") that a general rule exists. For example, a sign that says "parking prohibited on Sundays" (the exception) "proves" that parking is allowed on the other six days of the week (the rule). A more explicit phrasing might be "The exception that proves the existence of the rule."