Why Residual Graph Works

At max flow, there is no augmenting path from $s$ to $t$ in the residual graph.

This means $t$ is unreachable from $s$ using edges with remaining capacity. The vertices reachable from $s$ form set $S$. Since $t otin S$, you have $t \in T$. Edges crossing from $S$ to $T$ are saturated because they have no remaining capacity. These saturated edges are the bottleneck. Their total capacity equals the max flow. The max-flow min-cut theorem guarantees this.