Well, I don't really know much about it but I've had (and witnessed) many fitting sessions over the last 30 years of riding bikes, so here's the little I can tell you...

A basic fitting involves: 1-making sure that bike frame and components match your body measurements, posture, and any particular condition, 2-finding the optimal position for all components to literally 'fit' your body and riding style.
So first of all they take some measurements of your body, and then place the bike on a stationary trainer so you can test-pedal while they make all the necessary adjustments.

The first time you buy a bicycle the shop should take your measurements and give you a bike that is the correct size for you - i.e. the frame size must be correctly fitted to your body. Likewise, at the time of purchase, the shop makes sure that you have the right base components:
-saddle (sitbone measurements, padding/no padding, materials, etc)
-crankset (i.e. the crankarm length is right for you)
-handlebar (width, reach, drop are correc for your body and posture)
-stem (length, angle)

Once you have the correct equipment, a short fitting session (that should be done anyway at the time of purchase if you buy from a reputable shop) takes care of the following:
1-finding the correct saddle height
2-finding the correct fore/aft position of the saddle
3-figuring out whether you may need some tilt or you can ride a leveled saddle (most people can, some strange people like me cannot...)
4-finding out if you need any stem adjustments
5-finding the optimal position of your handlebar

A more lengthy (but much more useful) fitting process takes care of all of the above plus:
6-finding the optimal tension on your pedals
7-fitting the cleats in the correct position on your shoes. This is done with a powermeter on a stationary trainer, to find the position where there is no strain on the ankle and knee, and at the same time you can exert maximum power while pedaling.
8-finding the optimal position of the brake levers/shifters on the handlebars.
9-moving and readjusting cables depending on your riding style (% of time on the hoods, tops, drops) so that the equipment does not get 'in your way'.
10-finding the optimal (most aero, less disturbing for your posture) position for accessories like waterbotle cages or cyclocomputer.
11-making you pedal for a while going through all your gears and reading the powermeter and finding out if there is any glitch that needs to be taken care of.

A fitting session usually does cost some money, unless you are buying a new bike or the pro fitter works for your team - but it is definitely worth every penny.