The standard bore diameters / gauges were established back in the earlier days (muzzle loading blackpowder shotguns). Given the hard cardboard or fiber wads available in that day it was necessary to make sure the wads fit tightly in the bore to obtain a tight seal, otherwise you got gas blowing by the shot column causing distorted patterns and lost velocity. When breech loading guns and smokeless cartridges became common the shot shells were still loaded with a similar hard wad under the shot load. The wad still needed to match the bore diameter for the same reasons.
When today’s plastic wads became commonly used (the 1960 – 1970 era) it was found that the wad and shot column would expand to match the bore diameter. An exact match of wad to bore diameter became less important to shot shell performance.
Gunsmith Stan Baker was one of the first to experiment to see what happens when the bore diameter was increased. Boring out barrels from the standard .729” (on the 12ga) to as big as .800” Baker claimed both increased velocity and better patterns.
The expected velocity increase comes from pushing the same weight of shot down a barrel than is now larger; the larger the bore the less pressure required to achieve the same velocity. Conversely, equal pressure (i.e.; powder charge) should result in greater velocity (maybe 30-50 fps) in a larger bore.
The “better” patterns were attributed to less shot column constriction (i.e.; damaged shot) while the shot charge traveled down the barrel.
More explanations from other sources:
From Browning:
All 12 and 20 gauge Citori shotguns feature back-bored barrels. These barrels have a bore diameter increased to its maximum specification. This reduces the friction of the shot charge against the barrel wall, resulting in an increase in shot velocity. Because there is less constriction or pressure from the forcing cones on a shot charge, there are fewer deformed pellets, this leads to more uniform patterns than with standard barrels.
http://en.wikipedia.org/wiki/Gauge_(bore_diameter)
Shotgun bores are commonly "overbored" or "backbored," meaning that most of the bore (from the forcing cone to the choke) is slightly larger than the value given by the formula. This is claimed to reduce felt recoil and improve patterning. The recoil reduction is due to the larger bore producing a slower acceleration of the shot, and the patterning improvements are due to the larger muzzle diameter for the same choke constriction, which results in less shot deformation. A 12-gauge shotgun, nominally 18.5 mm (0.73 in), can range from a tight 18.3 mm (0.72 in) to an extreme overbore of 20.3 mm (0.80 in). Some also claim an increased velocity with the overbored barrels, up to 15 m/s (49 ft/s), which is due to the larger swept volume of the overbored barrel. Once only found in expensive custom shotguns, overbored barrels are now becoming common in mass marketed guns. Aftermarket backboring is also commonly done to reduce the weight of the barrel, and move the center of mass backwards for a better balance. Factory overbored barrels generally are made with a larger outside diameter, and will not have this reduction in weight—though the factory barrels will be tougher, since they have a normal barrel wall thickness.
Firing slugs from overbored barrels can result in very inconsistent accuracy, as the slug may be incapable of obturating to fill the oversized bore.