Why is such a design necessary ?
The basic function of the extra-ocular muscles is to position the cornea in the correct position , so that the object being viewed will stimulate the retina, and an image be seen.
Clearly, up, down left and right movements are required to move the eyeball. However, if you think about the kind of shape the orbit has it becomes clear why it six, not four muscles are needed. Take a look at the diagrams below.

 
The orbit is a pyramid shaped structure- with its base at the posterior aspect. The lateral and medial wall make an angle of 45 degrees with each other and 22.5 degrees with the optical axis, in which the eye lies.