The larger the number the smaller the pellet size. The smaller the pellet, the more there are in a shell.

I've always been averse to simplistic understandable answers

To be precise, here's what is on

Wikipedia. At least the last sentence is understandable.

*The gauge of a firearm is a unit of measurement used to express the inner diameter (bore diameter) of the barrel. Gauge is determined from the weight of a solid sphere of lead that will fit the bore of the firearm, and is expressed as the multiplicative inverse of the sphere's weight as a fraction of a pound, e.g., a one-twelfth pound ball fits a 12-gauge bore. Thus there are twelve 12-gauge balls per pound, etc. *