You may have noticed that on some suits, the sleeves have different lining to the rest of the coat. It isn't because they just ran out of the original lining.
Back in the day, cloth was horribly thick (20oz to todays 11oz). Clothing was almost uncomfortably warm, partly by design- there was no central heating in most buildings, and partly because of the technology involved in spinning yarn and weaving cloth.
By using silk linings in the sleeves, they became easier to slip on and off and also allowed a little relief in the underarm area, (which has its own cooling system).
The silk was too flimsy and expensive to use for the entire coat so alpaca was used everywhere else. Since the two fabrics would never be exactly the same shade anyway, tailors could get a little creative.
There is a myth that Edward VII was responsible for the stripes being introduced.
DISCLAIMER: I've only heard this story twice and have seen no evidence to suggest it is true but
I think its funny so I'll share it anyway.
Apparently Edward VII was a particularly enthusiastic, if somewhat messy, eater which would often result in juices from meats running down his arm, staining his sleeves. Since he had such a reputation for being fashionable (he is credited with popularising turn ups, the tuxedo as we know it, the Prince of Wales check, and the unbuttoning of the bottom button on coats and waistcoats) people began pre-staining their own jackets to be like the glamour King. This then developed into the stripes being woven in.
In truth it is far more likely that tailor's used it as a way to identify their own suits, an early form of branding. Nowadays most Savile Row firms have their house sleeve lining and none of them really look like stains I'm afraid.