It's full of analog circuitry like integrators, comparators, op amps etc (note: it's made from discrete transistors, so you won't find an opamp chip in there). The "state" as you mentioned is maintained as a a voltage.
The velocity of the ball is stored as a voltage, and the position is produced by integrating that with an integrator. Collisions are found with a comparator, and connected to the circuitry that maintains the velocity.
The output to the oscilloscope is two voltages that control the position of the beam.
Analog electronics is one of those dying art forms where the number of people capable of doing things like this is quickly dwindling. It's no longer a viable way of building actual products except for a few special applications.
I find it a sad state of affairs that we, the mankind, have learned and developed skills and knowledge that we'll be collectively forgetting less than a century later.
/saying this as someone who considers analog a black though beautiful art
See the link on the form post for the schematics and some example readouts of the signals.
Seriously impressive.