Well, I don't know this specific application but I can tell you what the outputs of linprog() mean in general.
linprog() minimizes the linear functional f'*A subject to a constraint, which is A*x <=b
The output fval is the value of the linear functional with the minimizing x: fval = f'*x (where here x is the minimizing vector)
In this case, it looks like x is the minimizing flow vector and fval is the value of the flow using that minimizing vector.