Projective Geometry: Why is multiplication defined this way? I am trying to understand this new way of multiplying in projective geometry. 

Why is it defined like this? Also does this have anything to do with multiplication using a slide ruler? (The picture in the link shows that $4 \cdot 4 = 16$ and $ 4 \cdot 2  =8$. Every unit is a power of 2. Slide rulers were commonly used in the old days way before the use of a calculator.)

 A: Error in image
First off, your image looks wrong to me. In my opinion, you'd want the following elements (too lazy to draw an image just now):

*

*a line $g$ connecting $0$, $1$, $a$ and $b$

*a different line $h$ through $0$

*a point $p$ on $h$, perhaps directly above $1$

*a triangle $t$ connecting $1$, $a$ and $p$

*a similar triangle $b$, $a\cdot b$ and $p'$
In other words, the construction would be:

*

*choose $p$ arbitrarily

*$p'$ is the point where (the connection of $0$ and $p$) intersects (the line parallel to $1\vee p$ through $b$)

*$a\cdot b$ is the point where (the parallel to $a\vee p$ through $p'$) intersects (the line $0\vee 1$)

In that case, your image would represent the Euclidean version of multiplication, which is a special case of the projective version.
In your image, the triangle over $1$ and $b$ looks similar to the one over $b$ and $a\cdot b$, so it appears that you'd have constructed $b^2$ instead of $a\cdot b$.
http://www-m10.ma.tum.de/bin/view/MatheVital/GeoCal/GeoCal4x2a has applets of both the Euclidean and the projective view of these situations, although the text is in German.
First question
For your first question, I guess the simplest answer would be “because it works”. So what is the objective? Given a projective scale along a line, i.e. the points $0$, $1$, $a$, $b$ and $\infty$, one wants to construct the point $a\cdot b$ using only tools from projective incidence geometry, i.e. joins (lines connecting points) and meets (intersection points of lines). There aren't many configurations which can accomplish this, but the von-Staudt construction your immage suggests can accomplish this, as you can check for the euclidean case and generalize as all operations can be interpreted projectively.
Second question
I see only a slight connection to the slide ruler. Both use some isomorphism to make multiplication accessible in a geometric way. One uses logarithms to translate multiplication into addition, whereas the other transforms it to the algebra of projective geometry. Apart from that, I see no connection.
