There are three parts to learn -- the electronics (handled by an Arduino and a few servos), the mechanics (for this design, it's not too complicated -- you could make do with popsicle sticks and some screws if you had to), and the software. The tricky bit with the software is the math. The arms, when free to spin around their centers, can be considered circles. You'd want to learn the geometry of circles and, specifically, how to find the intersection of two circles. The joints on the arms are where the circles meet.

I'm working a new version of my mobile testing robot that looks very similar to this clock. Haven't posted the code, yet, but I did have to go back and re-learn some trig and circle geometry to make the simulator. http://youtu.be/qRzrm4zx82g