The phrase “as the crow flies” means “in a direct or straight line, without any turns or detours as when following a normal road.” This phrase clearly alludes to the ability of crows and other birds for that matter to fly directly from point A to point B, as opposed to how humans are forced to work their way around the terrain and other features or follow an established path. The reason “the crow” (and not any other bird) is specifically used in the expression has to do with a very old method of navigation. Before there were reliable compasses, sailors would keep crows in cages on their ships. If they were lost, they would release one crow, watch the direction in which it chose to fly, and then follow it in hopes that it would lead them back to land.
The phrase “as straight as the crow flies” was used by Charles Dickens in Oliver Twist, and that’s what made the expression popular.