A flyer can be something or someone that flies on a plane, but another meaning is a handbill or advertising sheet handed out to people on the street. Because it's designed to go far, it gained the name flyer.
As far back as the 15th Century, Middle English used the word flyer to refer to something that travels quickly through the air, generally a bird or projectile. The connection with a person came about later, when air travel became more common. If you are a frequent flyer, you travel a lot. If you frequently post flyers, maybe you're in a band. Note: it doesn't matter if you spell the word flyer or "flier," as both are correct, but don't use both in one piece of writing.