The original note was about economy: if a rifle hangs on the wall in act one it must be fired by act three. The instruction is to cut. That reading is the more useful one, because unused conspicuous elements do real damage. An audience allocates attention to them and the attention is never repaid.
It applies with full force in short forms. In a ninety-second film every object the camera dwells on is a promise. A held shot of a photograph that turns out to mean nothing costs you the audience’s next held shot too, because they have learned your emphasis is unreliable.
The exception is texture. A world can contain things that are just there, provided the piece has not pointed at them. The rule is about emphasis.
What is Chekhov’s gun?
The principle that anything conspicuously introduced must pay off, and that anything which does not pay off should be cut. It is primarily an argument for economy.
Does every detail have to pay off?
No. A world can contain incidental texture. The rule concerns things the piece has emphasised (a held shot, a named object), because emphasis is a promise and an unpaid promise costs you the audience’s attention next time.