The 125DC systems I've worked with are for the control and relaying of medium and high voltage circuit breakers.
These breakers range from 4160 to 500,000 volts. They are a bit different than your basic 120/240 unit, but they accomplish pretty much the same functions.
The actual circuit breaker is nothing more than a switch. It can be operated by hand, or closed and tripped from a remote location, often by a PLC.
Since they are just a switch, relays are needed to trip them in the event of an undesirable condition. There are instantaneous overcurrent, long-term overcurrent, ground-fault and others.
Since these breakers will kill power downstream, in order to close back in, they need some sort of a power supply that is energized when downstream power is dead. This is usually accomplished with batteries.
The relays also need continuous power, so they are supplied with DC as well.
A typical installation will have one or more battery racks (usually totaling 125 volts, or thereabouts) and one or more chargers. The chargers will supply power during normal operation, and when their AC supply is cut off, the batteries will take over.
One really important reason why control power to the breaker must always be present is because if there's no power, the breaker relays cannot trip the breaker in the event of a fault.
Of course, this is a simple explanation that covers very little of a complete medium or high voltage installation, but you get the idea.