Which parts of the proof are you having trouble with?
It seems that the author should emphasize that the line segment through B_1 has as its end points some points A' and C', which lie on the corresponding rays OA and OC and NOT on the sphere . Otherwise it's all nonsense.
Moreover it is not clear why the law of cosines should be applied here. It is applied to the triangle B O C'. In my opinion the law thus applied says nothing, because a priori we should have seen that B_1 O C is the degenerated trihedral angle O B B_1 C' when B and B_1 coincide. So the value of the angle B_1 O C' is the lower bound for the values of the angle B O C'. And this, as it seems to me, is not implied by the law of cosines.
If one wishes to use the law of cosines, then it should be applied to the triangle B_1 O C', which is, again, the limit case of B O C'.