From the equation we can see this due to the fact that the matrix is diagonal. When changing the value of the EFG for a specific direction it will only change according to the component corresponding to that direction. Example: Ez decreases with increasing z. This shows that the other direction do not influence this and thus will probably cancel each other if anything is there to cause an influence. Which shows axial symmetry.
From the picture we can see this easily. Axial symmetry occurs when there is circular symmetry around an axis. Taking the x-axis, we can easily see that there is circular symmetry and thus also axial symmetry.