A proof using affine differential geometry can be found in Exercises 14 and 16 of Volume 3, Chapter 4 of Spivak's _A Comprehensive Introduction to Differential Geometry_.
However, one can give an elementary argument just using basic facts about quadrics (see, for example, section 4 of Chapter 8 of my book _Abstract Algebra, A Geometric Approach_ ). The outline is simple: If $L_1, L_2, L_3$ are three lines in $\Bbb P^3$ in general position, then there is a unique quadric surface $S$ containing them. It follows that any line intersecting all three of them lies on $S$, and hence the conclusion.