From page3 of the slide, the second point claims that |f| = the sum of f(s,v) where s belongs to V = the sum of f(v,t), where v belongs to V. (sorry, I don't know how to type this in markdown.)
I don't understand why the equation holds.
From the title it sounds like you read the equation wrong. The vertexes v in the first sum are the vertexes adjacent to s. They are not the same as the vertexes v in second sum. Those are the ones adjacent to t.
The equation says the total flow out of s (sum of the flows on edges from s) equals the total flow into t.
This must be true since the total flow into every other vertex must equal the total flow out (that's the flow conservation constraint). So if there is flow on an edge out of s, then if it goes to a vertex that is not t, then it must flow again out of that vertex, etc., until it reaches t.
All of the flow out of s must eventually reach t, because t is the only vertex that can consume it.