Flow in a network: Conservation of flow definition
The short answer is $f(u,v)=-f(v,u)$.  Note that the first sum includes both vertices $w$ such that $(u,w) \in E$, as well as vertices $w$ such that $(u,w) \notin E$ but $(w,u) \in E$.  Now unpack the implications of the first sum, separating by these two cases, and I think you'll see what happens.

For a more lengthy explanation, this is covered in standard textbooks.  Make a trip to a library to check out a few textbooks to find a detailed derivation; or, there are even online algorithm texts.