I'll start this off with a disclaimer that the origin is bloody hard, if not impossible to find - it's an old proverb that is variously described by Google as "old English" and "ancient", which gives you some idea of how far back this.

Why it became a saying is easier to figure out. Cats have quite stunning survival skills and traits. The main event that this refers to is, as you say in your question, high falls. Cats have what is called a 'righting reflex' which allows them to turn their bodies while in free fall so that weight is evenly distributed across their four limbs(as opposed to humans who often land feet first - ouch). This has been corroborated by a study by a group of NYC veterinarians who found that cats chances of surviving a fall actually increased when the distance they fell passed a certain threshold(somewhere around seven stories).

Now obviously cats didn't have New York skyscrapers to fall from in the past, however, natural heights such as cliffs and trees, and smaller man-made structures like roofs(churches maybe?), walls, and towers in towns and cities of the past would provide ample opportunity for cats to be witnessed falling from the heights yet miraculously surviving.

Perhaps the saying originated with a particular cat being witness to have fallen eight times without effect, before succumbing to a ninth. Or a cat that was 'revived' somehow after a fall. Another important question would be how did the passing of one of the cat's lives get noticed? Was it presumed that when a cat fell it instantly lost one like in a video game, but without visible death(think Crash Bandicoot's apples)? Or was it a cat being unresponsive before returning to mortal pursuits?

Either way it's pretty clear that cats have an uncanny knack for landing right way up and this has been noted throughout history in the form of the proverb. The present day reference to surviving events we wouldn't expect them to is probably not as modern as you'd think either.