It exists in nature... Look up images related to the Fibbonacci Spiral online (you'll find the nautilus shell, sunflowers, and myriad plants and animals). It's all around us whether we think about it consciously or not... When composition is woven into photography, it takes the viewer on a journey. Think about it this way: How much more pleasure would be derived by eating a delicious meal rather than just looking at it on a plate? How about a beautiful sports car? A horse? Wouldn't it be a more visceral experience to drive the car or ride a horse? As photographers, we only have 2 dimensions to work with and effective composition is one of the keys at our disposal. It allows our images to elevate the user experience. While a viewer's heartstrings are tugged when a photo carries special meaning, music plays in their soul if you can engage their eyes by first drawing attention to your point of interest (usually also the brightest spot) and carry them along by way of a natural flow. They may not know why, but they know that they have been touched in a unique and personal way.
Clematis with a Lensbaby Velvet 56 in Deb Hinchey's garden
Allium in the Palmer Visitor Center Garden
Examine some of the photos you find most pleasing and see if composition plays any part!