Why $n-1$ Iterations

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 $\leq 1$ edge. After round $2$, shortest paths with $\leq 2$ edges.

After round $k$, shortest paths with $\leq k$ 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.