The value I'm interested in is how fast the bike goes when I'm turning the pedals at my minimum-grindy, comfortable-cruisy and maximum-spinny cadences.
Nowadays that's easy to determine, since the arithmatic is a doddle when you have a calculator. It's even easier with the web calculators like Sheldon's and the one PS linked to.
Back in the day, they didn't have them, so they used the value called gear inches because that is only has one division and one multiplication in it (both just involving integers) - easy to do with pencil and paper or even in your head if you're good at that kinda thing.
Front chainring's number of teeth, divide by back cog's number of teeth, and then multiply by diameter of wheel in inches.
Hey presto, you have a value that is directly proportional to how far the bike will go forward when the cranks turn, and therefore directly proportional to how fast the bike goes at the aforementioned cadences.
It's still perfectly sensible to use it today because not everyone's cadence is the same. Also, it's all directly proportional anyway; it could be based on a completely arbitary value and still be useful, so the way gear inches comes out in the humanly comfortable 20-130 range of numbers is a happy accident.
Note that wheel size is chewed up and spat out in neatly digested format by the gear inch equation - a Brompton and a Pashley Guv'nor will both go the same speed at the same cadence when at the same gear inches.
Pedant NB: the diameter of a tyre is rarely an integer, but meh.