Historically projective drawing arose when artists discovered the art of drawing on their canvas so that the eye would see things in the drawing positioned just as they do when the eye saw the same thing in nature. This required drawing lines that intersect to depict parallel lines. It is no wonder that this feat was unobvious.
Desargues said of two triangles: if lines thru corresponding vertices are concurrent then intersections of corresponding sides are collinear.
Mathematicians discovered a few very pretty geometry theorems that were made unpretty when all of the qualifications about lines not being parallel were added to make them valid Euclidean theorems. Adding the notion of a line at infinity where even parallel lines meet made the projective ideas much cleaner. The connection to projective drawing was immediately seen.

In the late 19th century projective axioms were found to make these theorems legitimate in their own right. Indeed the axioms and theorems were cleaner than Euclid’s which Hilbert had to clean up about the same time.

In 1953 I had a short course in Berkeley in projective geometry that was axiomatic, simple and in classical style. I wish remembered the axioms we used. There are today on the web many uses of “projective geometry” and they differ considerably. Some speak of ‘desargueian geometry’ as a result of proposing axioms too weak to prove his theorem. Perhaps there is application for or beauty in the weaker axioms but I have found neither.

At the risk of furthering the disarray I note here that Grassmann’s thoughts and observations on the subspaces of an n dimensional vector space is exactly isomorphic with 19th century projective geometry. Indeed the notion of reduced row echelon form for matrices provides a canonical way to specify any element of an n−1 dimensional projective space when one has nominated a basis for the vector space.

Considering that so many features of the reals, extend to fields (especially finite fields), I am tempted to say that “projective geometry” is about just what happens on a hyperplane thru a vector space that does not include the origin. Invertible linear transformations of the space correspond exactly to the projective transformations in the hyperplane, and conversely. Indeed this isomorphism makes it difficult for me to separate the two sets of ideas. See this for further development.