A two channel power amp (such as one used for stereo amplification or mono bi-amping) is two power amps in one box. Each power amp will have its own output with two connectors, one positive and one negative. On an amp with five-way lugs, banana plugs or similar outputs, it may look something like this:
[+] [-] [-] [+]
A single four-pole SpeakOn cable can carry both signals to a bi-amp capable cabinet with just one cable. Usually the two amps in a two-channel amp have the same rated output power. So, for example, you might have two channels that can put out 200 Watts RMS each into a certain impedance.
What bridged mono mode allows is using both power amps to drive a single load, which gives you approximately double the power of using just one amp. So in the above example, while "stereo" mode is two 200 Watt channels, bridged mono mode turns that into a single channel that can put about about 400 Watts RMS. In practice it's usually a bit less than the sum of the channels, maybe something like 380 Watts in this case.
The way it does that is by connecting the amps together in series internally. The labeling on the outputs doesn't change, but if it could it might change to this:
[-] ==== [+]
==== indicates the internal connection between the two poles. So where before there were two pairs of positive and negative connectors, now there is a single positive connection and a single negative one, and also two connectors that should not be connected to at all. They are internally connected to each other.
When wiring using banana plugs or spade lugs or something like that, you can just manually connect the wire ends to the correct terminals. With a SpeakOn cable, you can't change the internal wiring of the connector on the fly, so you have to use a cable that has been wired correctly internally. If you use a four-conductor cable, you risk the two connectors that should be connected to anything getting connected to something. That would be bad. So a two-conductor cable is the safe option.