How are crossing gates triggered to come down and the lights signaled
to turn on at railroad crossings? I would like to know as much
technical information on how this system works.

When train passes certain point, prior to intersection, automatic
relay gets activated and turns on gate mechanism, light, and sound
warning system. All of these gets triggered off when train passes a
certain point after the intesection.

Designs vary from railroad to railroad but, in general, the same principles apply:

Railway tracks are divided into sections, typically 400 to 800 metres
long (Quarter to half mile) either side of the crossing. These
sections may or may not be insulated from other parts of the railway
line, depending on the system.

A low voltage electric current is applied across the railway tracks at
regular intervals down the track.

Sensing currents are often AC, especially when DC electric traction
systems are used and to avoid the interference of naturally occurring
earth currents.

Passage of a train down the track effectively short circuits the
electric current applied to the rails.

The changes in the voltage and current flow both along the track and
across the track are monitored by electrical/electronic equipment.

Current will flow across the tracks between the two (or more) voltage
sensing points that the train lies between. Sensing points further
away will have less or no current flow and will be at higher
(differential) voltages than at the points where the train is between.
Thus the position of the train is identified by what senring points
are short circuited and the direction is determined by what order the
sensing points are short circuited by the moving train.

Electric/electronic logic works out if the train is approaching or
departing from the crossing by the sequence that sening points are
activated in, first turning on then turning off the crossing
equipment. Signalling equipment (red & green section lights, etc.)
work off the same system.