So many keep quoting this but I fail to see how exceptions prove rules. In fact they contribute to disproving whatever it is that they are an exception for.

I believe the original meaning is more like if there is an exception, there must be a rule (otherwise it wouldn't _be_ an exception/you wouldn't need an exception). It's been watered down a lot, and is often used to suggest that any counterexample makes a rule stronger (which is obviously not always the case -- enough counterexamples just proves the rule is wrong/doesn't exist).

People misuse it a lot. Sometimes they're implying that if you can only find a handful of counterexamples, then you're showing how strong the rule is overall. Sometimes they're just spewing nonsense.

The original meaning goes like this: an exception of "if it's Thursday, you're allowed to have a brownie" proves/implies a rule of "no brownies"