Residual Graph

The residual graph shows remaining capacity after you have assigned some flow. For each edge $(u,v)$ with flow $f(u,v)$ and capacity $c(u,v)$, the residual graph has:

$1.$ Forward edge $(u,v)$ with capacity $c(u,v) - f(u,v)$ (remaining capacity).

$2.$ Backward edge $(v,u)$ with capacity $f(u,v)$ (allows you to undo flow).

The backward edge is the core concept. It lets you reroute flow if you find a better path later. This is what makes flow algorithms work.