Why is the solid angle due to an external point charge zero
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As viewed from a point outside the surface, the closed surface has a far side and a near side. For example, for a cube there is a front face and a back face.
If there were a point charge situated where the viewer is, the flux from this charge goes into the enclosed volume across one surface and out of the enclosed volume across the other face. The total flux into or out of the volume is zero. Likewise when measuring solid angle the angle subtended by one of these surfaces is taken as being +ve and that subtended by the other is -ve. The front and back faces subtend the same angle, but these angles add up to zero
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