Suppose we have problem of maximizing some cost function $c'x$ subjecto to $Ax = b$ and $x \geq 0$. The optimal solution is given by a basis $B$ (subset of $A$'s columns) such that $$\mathbf{x}_B = B^{-1}b$$
I understand the theory behind the simplex algorithm, but I still cannot see why a variable is appearing in the $i$-th constraint after solving. In particular, why $$ (\mathbf{x}_B)_i = (B^{-1}b)_i$$ and not any other variable. Does this have any interpretation? (For instance, in max flow)