Why does staring at a single point make it easier to balance on one leg? If I'm trying to put a shoe on while standing with one leg, it's easier to maintain balance while staring at a single point (e.g., on the ground).

Why does staring at a single point make it easier to balance on one leg?
Vision and balance are linked
Balance is fluid:
Vision and balance are connected within the brain. Balance is not something that the brain determines, is learned, &quot;this way is up&quot; and never revisited- like this smell = coffee, this sound = a bird; it is a continual process of biofeedback.
Three aspects of balance: vestibular system, vision and proprioception

Balance is the result of several body systems working together: the visual system (eyes), vestibular system (ears) and proprioception (the body's sense of where it is in space). Degeneration or loss of function in any of these systems can lead to balance deficits

We use our surrounds as a frame of reference, for our internal spirit level.  Maintenance of an accurate spirit level  involves continual measurement and feedback between the visual and vestibular systems, they are connected via the vestibulo-ocular reflex. When vision remains fixed upon an object, balance is achieved by the eyes moving to  counteract   the body's movements. These processes are interdependent, happening simultaneously, as the neurons that are stimulated within the visual system causing the neurons within the vestibular system to fire and vice versa.
Or simply put; Movement  is detected by the vestibular system, and the eyes give a plane of reference, offering proprioception.
It could be likened to axes on a graph, and both dimensions (vision and vestibular) are needed to plot the point of proprioception.
Vestibulo-ocular reflex:

The vestibulo-ocular reflex (VOR) is a reflex eye movement that stabilizes images on the retina during head movement by producing an eye movement in the direction opposite to head movement, thus preserving the image on the center of the visual field. For example, when the head moves to the right, the eyes move to the left, and vice versa. Since slight head movement is present all the time, the VOR is very important for stabilizing vision: patients whose VOR is impaired find it difficult to read using print, because they cannot stabilize the eyes during small head tremors. The VOR does not depend on visual input and works even in total darkness or when the eyes are closed.


Image courtesy of http://en.wikipedia.org/wiki/File:Vestibulo-ocular_reflex_EN.svg
An integrated view of balance:

The vestibular system works with other sensorimotor systems in the body, such as the visual system and proprioceptors in muscles and joints to check and maintain the body’s position at rest or in motion. These systems enable individuals to maintain a steady focus on objects even when body position changes and to safely maneuver in the environment.
The visual system specifically interacts with the vestibular system via the vestibular-ocular reflex, which senses head movements and generates equal and opposite eye movements to maintain image stability on the retinal fovea. The visual system is thought to provide continuous feedback about any retinal slip and the need to recalibrate the vestibulo-ocular reflex. (UC Davis Health System)

As we read and move our heads, our gaze is held steadily and we are able to continue reading uninterrupted thanks to the VOR. Given any alteration within the visual system, results in an alteration in our vestibular system and vice versa; when attempting a task that requires difficulty in maintaining balance, if the vision remains steady, as in constant upon an object, the vestibular system is able to measure (or locate)  the body's  proprioception and make adjustments to keep the vestibular system in the required plane (like the spirit level being kept level). So the effect is, we have one constant, vision and two variables; vestibular and spatial awareness.
Given the fluid nature of how the brain achieves balance; if the eyes were flitting and the visual field was varying, we have three variables. Mathematically this becomes a more complex formula for the VOR to determine where the body is spatially.
Interestingly, uncorrected poor vision can also cause poor balance when the eyes are closed.

... visually impaired individuals and those with uncorrected refractive error — those who could benefit from glasses to achieve normal vision but don’t wear glasses — have a significantly greater risk of diminished balance with their eyes closed on a compliant, foam surface than individuals with normal vision. (UC Davis Health System)

The link between balance and vision is an area of ongoing research.

References:

The Human Balance System
GOOD BALANCE IS OFTEN TAKEN FOR GRANTED
Vestibular Disorders Association

Common Conditions/Symptoms Associated with Vision
Brain and Eye Connection
Vision Clinic

Queensland University of Technology
Faculty of Health
Vision, Balance and Falls

How Our Balance System Works
American Speech-Language-Hearing Association