Dropout adjusters are intended to fine tune and facilitate rear wheel installation. You use the adjusters to roughly establish the fore/aft position of the wheel then dial one in or out so that the rim is centred between the chain stays. The next time you reinstall the wheel you just pull it back so that it seats against the stops and it should be in the exact same position.

While you can do it without adjusters, they certainly speed up the process. This is especially important in race situations where the ability to quicky and precisely align the rear wheel while changing a flat, is critical.

The fore/aft position allows one to optimize the shifting performance. Moving the wheel in the dropouts affects the chain gap, which is the distance along the chain from the point it contacts the freewheel to the point it contacts the jockey (top) derailleur pulley. Modern derailleurs, chains and freewheels tend to perform better with short chain gaps while older derailleurs, chain and freewheels perform better with longer chain gaps.

The fore/aft postion can also used to fine tune the wheelbase. A longer wheelbase offer a more comfortable ride and better straight line tracking, while shorter wheelbases makes the handling quicker and stiffens up the rear triangle. At least that how it works in theory. Many cyclists cannot tell the difference.

Moving the rear wheel back all the way will also sometimes allow you to gain a couple extra teeth of maximum cog size, for a lower gear.

The long, horizontal dropout was invented during the days of single speeds, to allow the accommodation different sizes of rear cogs without removing or adding chain links. Once multiple cog freewheels and derailleurs came along this was no longer necessary, except in rare cases like the Campagnolo Cambio Corsa. Manufacturers retained the long dropouts but added the adjuster screws. With the arrival of indexed shifting, manufacturers started incorporating vertical dropouts which prevent fore/aft adjustment of wheel position. This prevents owners from inadvertanetly altering the chain gap, which is critical in the optimum performance of indexed shifting systems.

It's impossible to say exactly where in the dropouts you should situate the rear wheel without knowing your bicycle and preferences. Start in the middle and experiment to find what suits you best.