The phrase itself appears in the errata documents for various D&D 3.5 publications, including those books which were included in the System Reference Document that forms the basis for Pathfinder. In that context, it is referring to contradictions between different D&D 3.5 products.
However, the phrase is used more generally to refer to the d20 System’s status as an exception-based ruleset. What this means is that the basic rules define general truths, but those rules can and are broken in specific ways by specific things. Wizards of the Coast, of course, has enormous experience with exception-based rulesets since it is the foundation for Magic: The Gathering – the game has particular rules, but individual cards have special powers that, effectively, allow you to break those rules in particular ways.
Pathfinder is still an exception-based ruleset, even if no Paizo-published document calls it out as such. It is the only manner in which the rules can be understood; every specific thing in Pathfinder relies on the general rules to define all the things that aren’t being defined in this specific case.
For example, a rapier is a one-handed weapon, but has a special exception relating to its use with Weapon Finesse – the rapier does not define what it means to be a one-handed weapon, it relies on the general rule for one-handed weapons for that. It only defines its own special case, the exception regarding Weapon Finesse.
If specific did not beat general, the general rule that one-handed weapons cannot be used with Weapon Finesse would be absolute – the rapier could not give itself an exception. Instead, the exception would have to be baked into the general rule – it would have to change from “one-handed weapons cannot be used with Weapon Finesse” to “one-handed weapons aside from the rapier cannot be used with Weapon Finesse.” And then the general rule would be responsible for listing every exception, which is, of course, impossible when you consider that some of the exceptions may not have even been written yet. Paizo would have to update the general rule for Weapon Finesse for every exception they wanted to print.
On the flip-side, if the rules were not exception-based to begin with, the rapier would be responsible for defining everything about its use – including those things it shares with every other weapon. That would get absurdly repetitive, since every other weapon would also have to define those things.