How do hairs know they've been cut?

They don't. All hair is deciduous: the hair grows for a while, stops, falls out, and then re-grows. The length of the "while" varies considerably; the hair on your arms, legs, and eyebrows has a fairly short lifetime, but the hair on the top of your head can last for years. Cutting any of it has no effect on growth. If you shave the fur on the tummy of your cat, you note that it grows back in a few weeks to look like the rest of the hair, and does not appear to change thereafter.