that puzzled me a lot, indeed. But the explanation is so easy to seem obvious: roughly speaking, I would explain this way:
Suppose that you are working in the plane - the usual plane, the plane you can imagine , the plane of the blackboard prolonged to the infinity - the so called "affine plane".
Consider there two parallel lines: don't worry, they don't meet.

But,if you consider the so called "projective plane" , obtained from the above plane by adding more "points", each one representing a "direction", the two lines will have the same "point",or, in other words, they have the same direction.