Why does this find Shortest Path?

BFS guarantees the first path to a node is the shortest. Here is why. Suppose the shortest path to node $v$ has length $3$. BFS processes nodes at distance $0$ first, then distance $1$, then distance $2$, then distance $3$. When BFS reaches distance $3$, it visits $v$. All nodes at distance $0$, $1$, and $2$ are already visited.

So the path to $v$ must go through nodes at those distances. Since BFS processes nodes in order of increasing distance, the first time you reach $v$ must be via the shortest path. There is no shorter path hiding somewhere else.