It represents each circle as an equation that is only true when x and y are on the circle. By requiring that all three equations are true, you can find all points that are on all three circles.
You can either convince yourself that three circles can only intersect at one point or you can use the fact that two variables and three independent equations means that there are zero or one solutions that satisfy all equations.
You could actually even make a system that only needs two distances (and the depth)! Two circles can only intersect at two points, so you just need to figure out which one of the two you are. That can be done by looking at which of the landmarks is on the left when looking towards them.
Now the really difficult thing here is to figure out why this works even with inaccurate inputs, as the math presented on the site assumes that everything is perfectly accurate.
You can actually formulate different ways of computing the position that differ in how they react to measurement error. One way to investigate that is to take the derivative wrt. to one of the radii.
This resonated with me because I once did the same thing but in 3d and with magnetic field strength instead of distance. I never found a satisfying solution because magnetic fields are capsule-shaped rather that spherical. The shape is described by a 4th degree equation, so its exact solution is too large to be useful and the whole system of equations cannot be solved symbolically.