What seems like a surprising fact (Desargue's Theorem) becomes intuitively obvious once you don't restrict the points to a plane. The three points a, b and c determine a plane and similarly for a', b' and c'. The line ab and the line a'b' are confined to their respective planes and thus their intersection must lie on both of these planes. Similarly for the intersection of lines bc and b'c', and that of ac and a'c'. So we now have the three points of intersection lying simultaneously in 2 intersecting planes. Since 2 planes intersect in a line, the intersections are collinear. Q.E.D.
Exercise for the student: Where in the above did I use the concurrency of aa', bb', and cc'?