Stable objects have both a broad base of support and a low centre of gravity. The centre of gravity of an object could also be called its balance point. If you support the centre of gravity, the object will balance and be stable. If an object is not supported directly below its centre of gravity then the object will be unstable and topple over.
Any object is most stable when the centre of gravity is near the centre of the base of support. Most structures, whether they are buildings or people, fall down because they are unstable, not because they are weak.
We can make objects more stable just by arranging them in a particular way. Objects with a small base are less stable than objects with a large base. For example, a pyramid is much more stable sitting on its broad base than on its point.
For any object that has a supporting base, its centre of gravity must be located directly over the base or it will tip over. This explains why a wide stance helps you keep your balance.
In order for you to stand up straight and not fall over, your centre of gravity must be directly above your base. Your base is your feet. As you move your centre of gravity (near your belly button) to and fro, it stays above your support (your feet).
When your feet are together, you cannot lean very far to the left or right since your centre of gravity won't be supported. When your feet are apart, you can lean further to the left or right since your base is much wider, supporting the shifting centre of gravity above. When your centre of gravity is no longer above your feet, you topple over.
A tall object with a high centre of gravity and a skinny support will fall easily, because even a little sideways motion pushes the centre of gravity off the skinny support.
For a fun extension: defy gravity with balance! Watch the video that explains this bottle rack!