Why c_f (u, v) = f (v, u) if f (v, u) not in E?
There is a significant typo on that slide. 

"$c_f (u, v) = f (v, u)$ if $f (v, u)$ not in $E$" should have been "$c_f (u, v) = f (v, u)$ if $(u, v)$ not in $E$" or, what is equivalent, "$c_f (u, v) = f (v, u)$ if $(v, u)$ is in $E$".




  Why do we care about edge that is not in $E$?


In fact, we care only edge $(u,v)$ if either $(u,v)$ is in $E$ or $(v,u)$ is in $E$. If $(v,u)$ is in $E$, it appears that we care about $(u,v)$, an edge that is not in $E$.

The reason can be seen by looking at the specific example in those slides written by Vincent Maccio.



The flow in the above network is not the maximum flow, 6. However, there is no path from $s$ to $t$. How about adding a flow of 2 along the edges along the edges $s\to 2\to1\to3\to t$? This flow is apparently invalid since the only capacity between 1 and 2 are from 1 to 2 instead of from 2 to 1. However, if we consider the net effect of all flows on the edge between 1 and 2, it is effectively 0 as we had a flow of 2 from 1 to 2. So the combinations of all flows, including the apparently illegal flow are in fact valid.

Now the problem is how we can find that apparently invalid but actually valid and helpful additional flow. We did it by a stroke of luck or a flash of intelligence. The systemic way is to adjust the remaining capacities on possible more edges. In the example, we can add an edge from 2 to 1 of capacity 2, which will enable the algorithmic search for additional valid flow.  That is, since directed edge (2,1) is not in $E$ (the directed edge (1,2) is in $E$), we would like to add edge (2,1) with capacity the flow through (1,2). That is " Why $c_f (u, v) = f (v, u)$ if $(u, v)$ not in $E$."



For more information, check the slides and related materials on the web.