How to interpret a line equation in 4-point geometry (affine plane of order 2).

So, now the base field, i.e. the number line instead of all the real numbers, consists only of $0$ and $1$, with $1+1=0$.

It is stated that the plane over this $2$ element field has six lines. Later those equations are just the equations of these 6 lines.

For instance, we have $A=(0,0)$ and $B=(0,1)$, and the line $AB$ is the one with equation $$X=0$$ and indeed, these two points satisfy this equation.
Or, take $AD$ with $D=(1,1)$, both points satisfy $X+Y=0$.
And so on.

The equations describe the lines, that is, if a point satisfies the equation then it is on the line. For example the line $X = 0$ is satisfied by all points $(X,Y)$ with $X = 0$ (with the coordinates $X$ and $Y$ coming from $\mathbb{F}_{2}$, the finite field of order 2). This gives an algebraic way to describe the lines.

You can think of the lines as having a slope of either $0$, $1$, or $\infty$ ($Y$ coefficient divided by $X$ coefficient), these slopes determine your parallel classes. This should let you go from a pair of points to the equation by determining a slope and then using a point to get the remaining value to describe the line as $AX+BY=C$.