What is the physics explanation for why an object with a larger base is more "stable" or and more able to "balance"?

With a large base, the force from the supporting surface can shift further away from a point directly below the center of gravity when necessary to oppose any external tipping torque. Any good physics text has a section on “Statics”.

It has to do with angular force balance [torque = R * force(R)] around the center of mass of a rigid body.

Those objects that have wider bases have, by definition, larger R at the base to compensate for external torque applied.

Those objects that have narrower bases have smaller R at the base to compensate for torque applied, even torque applied by the body's mass itself once the body's center of mass has moved beyond R (the "tipping point").

    _____________________
   /                     
  /                       
 /            c            
(         wide mass         )
              o------------->R
=============================

   _______________________
                        /
                       /
             c--->    /
                     /
                    /
                   /
                  /
           (narrow)
              o--->R
=============================

where c is center of mass.

The center of mass is lower in the body with wider base, so it cannot move as much to higher R (to "tipping point") as narrow body can when body pivots on base.