In a graph with n nodes, the longest simple path (no repeated nodes) has n−1 edges. You cannot have a simple path with n or more edges without repeating a node, which creates a cycle. After round 1, you have shortest paths with ≤1 edge. After round 2, shortest paths with ≤2 edges.
After round k, shortest paths with ≤k edges. After n−1 rounds, you have checked all possible simple paths. If no negative cycle exists, distances stop changing because you have found the true shortest paths. If distances still change, a negative cycle exists.