Why does an admissible heuristic mean A* is optimal?
With an admissible heuristic

The heuristic defines which nodes will be explored first, but does not change the final path found.

In your example, the heuristic will cause the path to Z to be explored first.  The algorithm will discover the true (expensive) cost of 120.  Then it will decide to explore the A path and discover the optimal route.

A better heuristic would guess A was the way to go immediately, find the cost of 100, and never bother exploring Z at all (e.g. if the heuristic said the cost was greater than 100).

So the heuristic changes the execution time, but not the final answer.

With an unadmissable heuristic

If the heuristic marked A as costing 130 (when the true cost is 100), it may find the route to Z (true cost 120), and instantly stop (because it "knows" the cost to A is at least 130), so return the wrong answer.