Draw the circle passing through $B,C,D$ and the circle passing through $A,C,D$ (or the line for three collinear points). The argument of the cross-ratio $\arg(A,B,C,D)$ is the angle between the two circles* where they meet at $C$.
To work this out, notice that the construction and the answer you get are invariant under Möbius maps, so you can make everything simple before you calculate by putting $D$ at $\infty$, $C$ at $0$, and $B$ at $1$.
To work out what the modulus of the cross-ratio should be, just swap some of the points over and repeat. For example, $(A,C,B,D)=1-(A,B,C,D)$ so $\arg[1-(A,B,C,D)]$ gives the angle at $B$ between the circles passing through $A,B,D$ and $B,C,D$.
There are several other angles that you could measure, but they are all related by various bits of spherical geometry (I find it easiest to think about this stuff on a sphere, by stereographic projection) in such a way that knowing two of them tells you all the rest.
$\ast$ There's some ambiguity in which angle to take, since there are two choices, and what counts as positive or negative. This can be resolved by checking for each circle the order the three marked points come in, and whether the fourth point is inside or outside the circle; details left as an exercise for the reader (translation: I can't be bothered to write them down).