The poem is a tiny argument about how things degrade with use. It starts with the feet, which "too soon" settle on "one ugly song to sing." That could mean a walker's repeated, tired steps, or the way any habit hardens into a single grim tune. Either way, the speaker sounds a little weary, as if variety has been lost faster than expected.
Then the view widens: "all hands in the world / grow dangerous with wear." Hands, which usually suggest making or comforting, turn threatening as they age and harden. The tone is blunt and a bit bleak.
The last lines break the pattern. "This is not true / of the moon." The moon stays untouched by handling and repetition, so it becomes a quiet exception. The title, "Proving the rule," plays on the old saying that an exception proves the rule: the moon's difference confirms how much the rest of us wear down. The mood is dry, resigned, and slightly wistful.