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Exceptions prove rules exist
The phrase sounds backward, because an exception looks like it breaks a rule. In one use, being memorable as an exception shows how normal the opposite is. Picture someone saying it's quiet in the country, and someone answering that two years ago a concert one night was so loud. Remembering that one loud night shows the country is, as a rule, always quiet.
The second use is practical: a notice of an exception tells you the rule exists even if nobody has stated it. A shop sign reading that all refrigerator orders must prepay for delivery means you don't have to prepay for delivery on anything else. The exception was made against a rule that actually exists, and the rule is proved by its exception.
Where does this phrase come from?
It comes from an old Latin legal maxim, exceptio probat regulam in casibus non exceptis, meaning that an exception confirms the rule in the cases not excepted. Lawyers used it to argue that a stated exception implies a general rule behind it. The English phrase was in use by the seventeenth century.
Why did prove once mean test?
The verb prove comes from the Latin probare, which meant to test or to show to be good. That older sense survives in proving ground and in the saying that the proof of the pudding is in the eating. Alcohol proof kept it too: the term comes from an old English test in which spirit was poured on gunpowder to see whether it still burned.
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