There isn't anything magical about developing a photographer's eye. There's a lot to be said for observing as much as you can about the world around you and appreciating the wealth of details. But when it comes to still photography, you need to "edit" your world view by placing four walls or a frame around your subject, cutting off the rest of the world and asking your audience to focus on a specific part of what you've seen.
The Rule of Thirds is a good way to help you with composing what your photographer's eye spots. Your goal is to create a final image - bordered by a frame - that's both dynamic and pleasing. The following video from UK photographer and teacher Mike Browne runs you through the rule as well as a few basic examples. If you're not already using this easy way to compose your shots, try it out and you'll see the difference between applying the rule to your subject ... versus a static composition with your subject completely centered.