For instance, given a bike and a rider, one can adjust seat height, seat position, stem length, handlebarhheight, crank arm length and possible more things to get the correct reach and stem height. However, all that also has to be adjusted to account for body size (limb lengths), mobility etc, in order to level the hips, get the knees correctly over the pedals, aero position, correct weight on the hands, no chafing in the butt etc. It's really an art to adjust everything at once, as fixing one thing affects the others.
So while a bikefit can cost about $300, it's money well spent if one is serious about the biking (and one has then probably spent ~20x that on bike+gear anyways.. ). I'm about to head out for a ~120km ride today, so I've found a bike fit invaluable.
My friends and I are avid mountain bikers, and semi-frequent road bikers and most of us have a pretty decent grasp on the effect of the main factors that will affect how a bike rides and handles and discussions are often had on the validity and suitability of different bikes geo choices for different kinds of riding. None of us are experts and we couldn’t tell you exactly how a bike would feel just by looking at the geometry, but you get a decent mental model after a while.
Granted, different people are enthusiasts for different reasons, for instance I rely on my bike for getting around town, but will never be a competitive rider. So, the interest in geometry ranges from just being safe and comfortable to ride, up to performance aspects of activities such as mountain biking, racing, or touring.
One hassle is that different makers sometimes supply a different set of measurements, especially inexpensive or older bikes, so you have to figure things out a bit. Also, a short test ride might not tell you how happy you will be after 100 miles. If there are things that you are happy or unhappy about on your existing bike, I suggest measuring it and try to figure some things out that way. When shopping for a bike (or hunting through piles of used bikes, as I've done), bring a tape measure.
http://cycle-ergo.com/ is a site some motorcyclists use to figure out which bike suits their body and preferred riding position.
some bikes even offer adjustable geometries by moving a bolt to a different position, which really makes it easy to understand how a change in geometry affects the handling characteristics.