If a plate is elongated in the x and y direction, why is the stress sigma z considered negligible with respect to sigma x and sigma y?
Because along z direction, body thickness is so small (ideally zero) that practically it consists of only the two surface boundaries (top and bottom, see figure) . Because no pressure is allowed in these boundaries (otherwise we violate plane stress conditions), equillibrium of internal stresses and external actions cant't be satisfied on them, unless $\sigma_z = 0$. Because no material exists (ideally) between these two boundaries, no $\sigma_z$ is possible "internally".
Typical applications of plane stress conditions include thin shells, loaded only in their plane.