The basic trick of this robot is high school trigonometry.
If you know the distance between the steppers and you know the length of the strings then you can calculate the XY position of the pen. If you know the XY position of the pen and the distance between the steppers you can find the length of the strings. From there you can figure out how much the strings have to change to move from one place to another and then you can draw lines and curves.
The rest is boring tricky stuff, nice but not vital. There's some code to make sure you can't draw outside the drawing area, and some code uses trapezoidal motion interpolation
to draw accurate pictures faster.