They're showing a really old tower in that video. Most of the system does not use mechanical lever interlocking machines. It's mostly General Railway Signal NX, which is entirely relay-based.
NX, (for eNtry eXit), is probably the first system with an intelligent user interface. The dispatcher has a track map, with lights showing which blocks are occupied. To route a train through the controlled area, the dispatcher pushes a button on the track map at the current location of the train. Lights then come on for all the places the train can reach from there. This takes into account other trains present, other routes already set up, and track or switches locked out for maintenance. The dispatcher presses an exit button where the train is to go, which locks in the route. All the signals and switches are automatically set by that one button press, and conflicting changes and routes are locked out. As the train progresses through the control area, the switches and track behind it are released, and can be used for other trains.
It's designed to be fail-safe. There are "train stop" devices at every signal. These raise a big metal lever at the right of the rails. On the lead truck of each subway car, there's a big metal lever connected directly to an air valve for the air brakes. If a train passes a red signal, the air valve trip lever hits the train stop, which slams on the brakes and cuts power. The signal system even checks the position of the train stop devices; there are always at least two train stop devices up and locked between two trains, and train stops can't clear until the next one is up and locked.
It's a good system. The main problem is the sheer number of relays and the amount of wiring required.
Train detection is by checking if the rails are shorted together by wheels. In keeping with the fail safe concept, the power for that check is fed into one end of the block, and it's sensed at the other end. Thus, a rail break or a power supply failure causes the system to sense a train and set signals red. On electric railroads, the rails are also the power return for traction power.
(The London system, with two power rails, is an exception.) So the sensing signals have to be different from traction power. They're usually audio frequency signals, with a different frequency used for adjacent blocks. At block ends, there's a low-pass filter capable of handling a few thousand amps, basically a big inductor. One of those failed in DC, causing a crash.
Washington, DC has a similar system, but on top of it, they have a computer-controlled system for dispatching. This is quite common - a dumb, reliable system based on trackside equipment for safety, and a computer-controlled system for scheduling and dispatching.
All this stuff has to handle snow, ice, snowplows, flooding, the huge traction currents (600VDC in New York), and lightning strikes on the rails as part of normal operation. So everything is in hulking iron and steel cases and very rugged.
There's a relay-level simulator for NX.[1] Runs on Windows.
[1] http://www.nycsubway.org/wiki/NXSYS,_Signalling_and_Interloc...