To each of our 23 people, we can assign a number in \(\{1,2,\cdots,22\}\), representing how many positions clockwise the lazy susan needs to be rotated in order to align that person with their dish. Notice that this number is never 0, since the dishes were completely mismatched. So there are no more than 22 possible rotation values.
We have 23 people matched to 22 or fewer rotation values. So, by the Pigeonhole Principle there are at least two people with the same rotation value. So we can rotate the lazy Susan so that both of them are matched with their correct dish.
Once you see the trick, the proof suddenly becomes obvious. It's just finding that trick ...
Check that your proof speaks about people, dishes, and rotations of the lazy Susan. If it's all equations, rewrite it so that it addresses the practical question in the original word problem.