Imagine water flowing through pipes. The maximum water you can push from to is limited by the narrowest chokepoint. You cannot exceed that limit. That chokepoint is exactly the minimum cut. You cannot push more flow than the cut capacity, and you can achieve flow equal to it.
The cut is the limiting factor. Once you reach max flow, there is a cut where all edges are fully used (saturated). That is your minimum cut. Flow backs up at this bottleneck.