As an example let us show that any two lines, $L_1$ and $L_2$, of your projective plane intersect at exactly one point.
If one of them, say $L_1$, consists of the points at infinity only, then $L_2$ intersects the affine plane along a line $L_2'$. Therefore $L_2$ contains exactly one infinite point, namely the one associated with all the affine lines parallel to $L_2'$. Therefore $L_1\cap L_2$ is
If both $L_1$ and $L_2$ intersect the affine plane (along the lines $L_1'$ and $L_2'$), then
there are two possibilities. If $L_1'$ and $L_2'$ are not parallel, then they intersect at
a finite point. The same holds then for $L_1$ and $L_2$, as the infinite points are distinct (due to them not being parallel). OTOH, if $L_1'$ and $L_2$' are parallel, then they don't have any common affine points. But then the same point at infinity is tagged on them to get the lines $L_1$ and $L_2$, so the projective lines intersect at an infinite point.
The other axioms of a projective plane are verified in the same way. For example, you
show that there is a line through any pair of points by splitting it into separate cases
according to whether neither, one, or both of the points are infinite or affine.