MST - algorithm to add an edge to the graph
Let $T$ be the MST of the original graph $G$ and let $e'$ be the newly added edge.
Suppose for simplicity that all edge weights are unique (if they are not, fix any arbitrary linear order among $E(G)$ that prioritizes the edges in $E(T)$. Then break weight ties according to this order).

Let $e$ be the edge of maximum weight in the unique cycle $C$ of $T + e'$. And assume that $e \neq e'$ (otherwise the proof is trivial).

One way to prove that $T-e+e'$ is a MST of $G+e'$ is showing that no edge $f \in \{e\} \cup E(G) \setminus E(T)$ can belong to any MST of $G+e'$ using the cycle rule.

If $f=e$, $f$ cannot belong to any MST of $G + e'$ because it is the edge of maximum weight of the cycle $C \subseteq T+e' \subseteq G + e'$.

If $f \in E(G) \setminus E(T)$ then, since $f \not\in E(T)$, $f$ must be the edge of maximum weight of some cycle $C' \subseteq G \subset G + e'$ and hence it cannot belong to any MST of $G + e'$.