Square format images appear to be so charming, in part, because of the composition of the image. Since all of the sides of the image are the same length, the image can be broken down into nine equal squares when the rule of thirds is applied.
If you compose your scene with your main subject on one of those lines, it will always be just slightly off center—enough to make the image that much more charming and appealing. To be fair, some love square format so much that they may argue that you can throw the rule of thirds out the window.
The Way the Human Eyes Work
In photography school, I was often taught about how the human eye moves from left to right when looking at an image. This mostly applies to 3:2 aspect ratios. However, with a square format image the eyes have less to scan than when they look at a rectangular image. Indeed, everything in the scene is mostly centered, so the eye tends to stay in one general area, and therefore has less to internalize.
And therefore the eyes will either scan the photo in a circular fashion or start towards the center and work their way around.
Of course, good subject matter is also key to a great image.
More Packed into a Scene
Adding onto the way that the human eye works, a square format image has a more natural sense of balance due to the equilateral lines, as opposed to a rectangle.
One reason for square format’s popularity amongst many photogs was the fact that you didn’t need to do any cropping later on. It’s experienced a revival in mobile apps and with digital camera aspect ratios for the same reason and the sense of balance.
If your camera only takes rectangular photos, you can give your images a square crop in a program like Photoshop or Lightroom. You’ll see that this can not only make your images look simpler, but also much better, by cutting out lots of potential clutter.