What did one plate say to the other plate?
Why this joke works
A Plate Offers Dinner
Dinner is on me normally means the speaker will pay for the meal. A plate can say the same thing because the food literally rests on its surface. Giving tableware a speaking role converts an offer of generosity into a statement of location. Read the reply in a friendly host's voice. No extra restaurant scene is needed because dinner and the plate already establish both meanings.
A closer look
Why the plates need a conversation
The joke needs one plate to address another, even though either plate could carry food. That tiny social scene makes the line sound like an invitation between people. If the setup only asked where dinner was, the answer would be a plain description. The conversation is what makes the payment interpretation arrive before the physical one.
Fact behind the joke
A small word with several jobs
The preposition on can express physical support as well as responsibility for an expense. English uses the same short word for both relationships. A dinner invitation activates the expense sense, while a serving plate gives the support sense a visible everyday example.