Mooncoon hits it right on the head. The short answer is: Before the advent of smokeless powder and metal jacketed bullets, paper patching of bullets resulted from the need to reduce leading and black powder fouling in cartridge rifles and improve their reliability in the field. From a military standpoint, this was vitally important. Also, to allow higher bullet velocities. A paper patch fulfills the same purpose of today's metal-jacketed bullets...to reduce lead fouling in the bore. Why paper patch? As Mooncoon mentions, sometimes this is necessary, where a particular firearm's rifling depth won't allow the use of naked lead bullets. The other, is to custom fit bullets to those rifles with way oversized bores that would be impractical to use otherwise. As an example: Some .303 Lee Enfield service rifles may feature bores well in excess of the nominal .311-.312" bore diameter; often measuring .315" or more. Obviously, a standard diameter .311-.312" metal jacketed bullet will just rattle down such an oversized bore without touching the rifling. The result: non-existent accuracy and yet another perfectly serviceable rifle relegated to becoming a wallhanger. But, where a paper-patched bullet is substituted that more closely matches actual bore diameter....the venerable .303 rifle that served the Empire so brilliantly for well over a century, once again regains its proper and respected place in the hunting field or target range. It bears mentioning that bore tolerances in today's modern firearms don't seem to suffer from the variation often seen a century or more ago. But, there are still plenty of those old boomers still in use today that do. So, paper patching becomes a viable option.
As to when, where and whom started paper patching bullets, this is unclear. Almost certainly, the practise evolved from the use of cloth patching rifled musket balls. To reduce also, the dreadul lead fouling experienced in many early rifled gun barrels, where naked lead bullets were used. Paper patching was also popular amongst 19th Century long range marksmen. Very much for the same reason it is today. The paper patch provides greater shot to shot consistency and thereby accuracy, than is capable of many lubricated naked lead bullets.
One question sure to arise from this forum's discussion so far, is: Can paper patched bullets be used with smokeless powder? Sure can. In fact, paper patched lead bullets can be driven at or very close to the standard muzzle velocity of many of today's cartridges. A bit more care may have to be taken with powder selection and chamber fit, but thats about it. An earlier post mentioned that paper patching may be too complicated to bother with. Not so. The fact is, most people can learn how to properly paper patch a bullet in well under an hour. After that, it just takes practise to perfect the technique. Yes, it is a bit more work than just shoving a jacketed bullet into a case. The real payoff is in the knowledge gained by the experience. Plus the direct link a shooter forges to the history of firearms development. Works for me. I suppose one way of looking at it is that as long as a person has a supply of paper and lead, they'll never run out of ammo.