Now that I see the context in which the argument appears, it makes sense.
If ${\bf x}^*$ is an optimal basic feasible solution with objective value $z^*$, and ${\bf x}$ is any other feasible solution with objective value $z$, then (from derivations earlier in the text), they say
$$z^* - z = \sum_{j \in J} (z_j - c_j) x_j,$$
where $J$ is the index set for the variables that are nonbasic for the optimal solution $x^*$ and not necessarily for any other basic solution. For the rest of their argument (and the one below), $J$ retains this interpretation.
The other piece of information they are relying on (and this is the point I think Robert Israel is trying to make in his answer) is that the values of the basic variables for any basic solution are obtained by setting the nonbasic variables to $0$ and then solving the resulting set of linear equations. (The simplex tableau makes this automatic for you, so that you can just read the solutions off of the right-hand side.) The point is that if $x_j = 0$ for all $j \in J$ then ${\bf x}^*$ is the solution you get. Thus if we have a feasible solution ${\bf x}$, distinct from ${\bf x}^*$, then there is at least one $x_j$ for $j \in J$ that is nonzero.
Since all the variables have to be nonnegative in order for ${\bf x}$ to be feasible, $x_j > 0$. With the assumption that $z_j - c_j < 0$ the equation above forces $z^* < z$. Thus any other feasible solution has a strictly larger objective function value and so cannot be optimal.