Is Dijkstra's algorithm just BFS with a priority queue?
You can implement Dijkstra's algorithm as BFS with a priority queue (though it's not the only implementation).

Dijkstra's algorithm relies on the property that the shortest path from $s$ to $t$ is also the shortest path to any of the vertices along the path. This is exactly what BFS does.

Or in another perspective: how would Dijkstra's algorithm behave if all the weights were 1? Exactly like BFS.