A* expands the search tree by expanding the node for which the past cost ($d(n)$; cost of path from the start point to the node) plus the heuristic value ($h(n)$) is minimum. Because the heuristic $h(n)$ is admissible, $d(n)+h(n)$ is a lower bound on the cost of the path to the goal that may be obtained by expanding the node $n$.
If we have found a path of cost $L$, we can (only) conclude that path is shortest when there are no more nodes for which $d(n)+h(n)<L$. If there were such nodes then expanding them could possibly give a path of cost $<L$.
Assume the shortest path has cost $L$. Consider the graph, but restricted to the nodes for which $d(n)+h(n)<L$. Then the nodes that A* expands are exactly the nodes in the connected component of the start point in this restricted graph.
If we have an alternative heuristic function $h'(n)$ such that for all nodes $n$, $h'(n)\leq h(n)$ then clearly the size of the connected component containing the start point in the graph restricted to nodes where $d(n)+h'(n)<L$ is greater (or equal to) the size of the same connected component in the graph restricted to nodes where $d(n)+h(n)<L$ because $d(n)+h'(n)\leq d(n)+h(n)$ for all nodes $n$.
This analysis assumes that there are no other nodes (other than the goal node) for which $d(n)+h(n)=L$, if not this is the case then how many nodes are expanded depends on the tie-breaking order (and we can not say anything useful about that).