A phase inversion is exactly that. We flip the phase of one pickup to achieve a cancellation effect.  Basically what that means is the + and - wires are switched on 1 pickup only. In the case of the Jimmy Page harness, it's the bridge pickup.  Every guitar pickup produces a waveform that can be viewed on an oscilloscope, and they look very much like waves in the ocean.  It is a matter of physics that any two waves equal and opposite will cancel each other out and equal "zero" again, just like waves in the ocean:  If you have a wave moving away from shore and another toward the shore of equal size and strength, when they meet there will be zero energy.  The same is true for electrical signals.  An "out of phase" sound occurs when BOTH pickups are on and the electrical signal of 1 pickup is inverted.  Since the neck produces some sounds and harmonics that the bridge pickup cant, and vice versa the result is a low powered, kind of tinny but very harmonic rich sound.  The fundamental or main note cancels leaving the lesser harmonics to be heard.  BOTH pickups must be on (3 way switch in the middle) while the phase inverter is active to hear the effect.

Secondly lets talk about the serial mode operation.  Serial mode links your two humbucker positions in series with one another.   This effectively doubles the power output of your guitar and gives more drive to your amplifier, but it does not do so without compromise.  A humbucker pickup is essentially two single coils wired in series which is why a humbucker is more powerful that a standard single coil, and also why they are darker sounding with more compression and treble-cut.  The same will be true when putting two humbuckers in series as they will be louder but also have more treble cut and compression.  When entering the serial mode on the Jimmy Page harness, the 3 way switch is bypassed and has no function as the nature of "serial operation" means that both pickup positions will be on as a default.  

Lastly, when serial mode and phase inversion mode are combined we get some really interesting effects.  Serial mode naturally turns both pickups on so using the phase inverter during serial mode produces a very prominent tonal change.  Both pickups will be active due to being in serial mode so the effect of the phase inverter is heard regardless of the position of the 3 way switch.  The more powerful mode of serial operation causes the harmonics produces by the phase cancellation to jump out of the amplifier creating that "Jimmy Page" sound.  Also noted for this type of configuration is artist "Brian May"  The result is a very powerful also harmonic rich tone with lots of treble and bite, kind of like a half cracked wah pedal sound.  By toggling the coil splits on and off during this mode the user is able to change which part of the string is read by the pickups and thusly alter the phase cancellation radically producing very vocal type changes in the sound.  

As a side note, when selecting the bridge pickup by itself, in single coil mode AND toggling the phase inverter switch back and forth, you are able to select one or the other of the bridge pickup coils resulting in two different single coil sounds in the bridge position.