Bellman-Ford relaxes every edge in the graph, then repeats this process multiple times. Each complete pass through all edges is called one iteration or one round. After rounds, you have found shortest paths using at most edges. This is because each round extends shortest paths by one more edge.
Since the shortest path between any two nodes uses at most edges (no repeated nodes), running rounds guarantees correct distances for all nodes. If no negative cycle exists, distances stabilize after rounds.