The given defect is called astigmatism. It arises because the curvature of the cornea plus eye lens refracting system is not the same in different planes.
The eye lens is usually a spherical, i.e., has the same curvature in different planes but the cornea is not spherical in case of an astigmatic eye.
In the present case, The curvature in the vertical plane is enough, so sharp images of vertical lines can be formed on the retina.
But, the curvature is insufficient in the horizontal plane. So, horizontal lines appear blurred. This defect can be corrected by using a cylindrical lens with its axis along the vertical.
Clearly, parallel rays in the vertical plane will suffer no extra refraction, but those in the horizontal plane can get the required extra convergence due to refraction by the curved surface of the cylindrical lens if the curvature of the cylindrical surface is chosen appropriately.