In the $A^*$ algorithm, the optimality of the path is guaranteed when the heuristic has the property of being admissible or monotone\consistent.
I was able to understand the admissible property, however, I am not able to wrap my head as how the monotone property guarantees that any node can be explored only once (i.e., the path that lead to that node has the shortest path). As stated in the wiki page ,
"If the heuristic $h$ satisfies the additional condition $h(x) \le d(x, y) + h(y)$ for every edge $(x, y)$ of the graph (where $d$ denotes the length of that edge), then $h$ is called monotone, or consistent. In such a case, $A^*$ can be implemented more efficiently—roughly speaking, no node needs to be processed more than once (see closed set below)—and $A^*$ is equivalent to running Dijkstra's algorithm with the reduced cost $d'(x, y) = d(x, y) + h(y) − h(x)$."