Why do I need to "ground" my guitar pickup to the guitar bridge
The 'ground' of the output jack is actually a shield (and probably connected to chassis ground). It prevents outside electric fields (especially the 60Hz hum) from capacitively coupling into the audio signals inside the cable and jack. By soldering the bridge to the shield, you are extending the shield around the pickups. 

Noise can couple into circuits capacitively (via electric fields) or inductively (via magnetic fields). Shielding stops electric fields. Its also good not to turn cabling into an antenna (which could have also been happening with your setup.

Magnetic fields are more difficult because they can only be attenuated by materials. The best thing to do is not create loops which make antennas and avoid mutual coupling (every wire with current creates a magnetic field)

An ideal shield looks like the bottom diagram, since you can't put a shield all the way around the pickups, grounding it is a close second.