The Core Idea

Floyd-Warshall's core insight: can I improve the path from $i$ to $j$ by routing through an intermediate vertex $k$?

If dist[i][k] + dist[k][j] < dist[i][j], then going through $k$ is better. The algorithm considers vertices one at a time as potential intermediates. After processing vertex $k$, all paths using vertices $0, 1, \ldots, k$ are optimal. This builds up the solution incrementally, which is the essence of dynamic programming.