Single shot firearms were the natural result of firearms development. When the metallic cartridge was invented and made practical, firearms were of the muzzleloading type. It only made sense to 1) convert some of those guns into cartridge guns in the simplest possible manner, and 2) use existing factory machinery and gun parts to make simple, easy-to-build guns. Both of those scenarios resulted in single shot firearms.
It wasn't too many years before the advent of guns with attached magazines or rotating cylinders rendered the single-shot concept obsolete for most guns, but it has held on over the years. In some cases, such as break-open shotguns, it is often a matter of affordability - they are cheap. Sometimes it's just a matter of liking the looks of an old-timey gun, or the challenge of trying to make a kill with only one shot when hunting.
I have always found that last goal to be a questionable reason to hunt with a single shot rifle. Yes, I strive to make clean, one-shot kills - but I carry a repeater whenever it's legal, because life is unpredictable and I want to have the ability to follow up quickly if need be.
Some single shot rifles and pistols are far more costly than a good bolt action repeater, which is another argument in favor of hunting with repeating firearms.
Single shot firearms will always have a place in some shooters' hearts, and that's as it should be. Even though single-barrel muzzleloading guns can only fire a single shot per loading, they are not usually referred to as "single shot." That term is normally reserved for use with cartridge-firing guns.