The first two of Maxwell's equations let us calculate the electric field magnitude and direction due to any static charge distribution.
Between the planes the field points from the positive towards the negative plane. Hence, if the volume in question has no charge within it, the net flow of Electric Flux out of that region is zero.
The flux of a electric field through a closed surface is always zero if there is no net charge in the volume enclosed by the surface. In general a positive flux is defined by these lines leaving the surface and negative flux by lines entering this surface.
The flux passing through the top, bottom, front, and back sides of the cube is zero since these sides are parallel to the field lines and thus do not intercept any of them.
If you use the water analogy again, positive charge gives rise to flow out of a volume - this means positive electric charge is like a source a faucet - pumping water into a region.
The field must therefore be parallel to the z-axis. In the next section, this will allow us to work with more complex systems.
Intuition trumps complication, always. The flux through a given surface can be positive or negative, since the cosine can be positive or negative. The density of the field lines is proportional to the strength of the field. They do not address the origin of the forces.