Why Two-BFS Works

Say the true diameter is the path from $x$ to $y$. When you BFS from a random node $s$, you will reach some farthest node $u$.

If $u$ is not on the diameter path, then there is a longer path from $s$ through the diameter to one of its endpoints. Contradiction: $u$ would not be farthest.

So $u$ must be on or reachable-to an endpoint of the diameter. When you BFS from $u$, you will reach the other endpoint, giving you the full diameter.

This is not a full formal proof, but it is the core intuition.