Why is it that the flow value can increased along an augmenting path $p$ in a residual network?
Instead of giving a formal proof I will present some intuition on why augmenting paths increase the flow. Look at the picture below. Let's go through some augmenting path, for example the one painted in black. It increases the flow by some $x$. Then go through another one (that has $c_f(p) = y$) that contains an edge which pulls back some flow (edge $(u, v)$ is present only in residual network). When the red path enters $u$, it increases the flow that comes into $u$ by $y$, but then pulls $y$ units of flow back to $v$, so $v$ can use edge from red path to push those $y$ units somewhere else. So you see that $y$ units came to $u$, $y$ units was withdrawn from $u$, $y$ units came back to $v$ and then $y$ units went through the red path. The property that income flow equals the one that goes out is preserved. Capacity constraints (flow through an edge is not greater than capacity of this edge) are also preserved, because $y$ is the smallest edge weight on the red path. As you can see, going through the red path increases overall flow by exactly $y$, because augmenting path  doesn't decrease the flow anywhere, it rather redirects it in the manner described above. It's like vertex $u$ saying to vertex $v$ "Oh hey, the red path gave me some flow, maybe you ($v$) would take back some of the one you gave me and send it to someone else?". I'd say that adding "redirecting" word somehow to "augmenting path" would clear some misunderstanding about this concepct. 

Having this intuition, I think that making a formal proof from it should be fairly easy.