Where is this line from? (I hope I did not write it!)
For a field $F$, we can think of affine $n$-space $A$ as sitting inside the vector space $F^{n+1}$ as the *affine* subspace given by $x_0=1$.
A line in $A$ is given by its intersection with a 2-dimensional vector subspace of $F^{n+1}$
If $P_1$ and $P_2$ are two such 2-dimensional vector subspaces, then the corresponding lines are parallel if the intersection of $P_1$ and $P_2$ is a line that does not meet $A$.
However, by our definition of projective space, each such line does give a point in projective space.
Thus we can see that lines that do not meet $A$ have been *added* to the affine space in order to get projective space.
Hope this clarifies.