After iterations, all shortest paths are finalized if no negative cycle exists. Any simple path has been fully explored. Run one more iteration (the -th round). If any distance improves, a negative cycle exists. Why?
Because improving a distance in the -th round means there is a path with or more edges that is better than the path with edges.
A path with edges in a graph with nodes must contain a cycle. If this path is shorter, the cycle must have negative total weight. You can traverse this cycle repeatedly to make the path arbitrarily short.