How to answer the question: How does a bicarbonate buffer solution control pH when either an acid or a base is added?

A buffer solution is resistant to changes in pH upon the addition of relatively small amounts of acid or base. In this case, the equilibrium resisting changes to the pH is: H2CO3 <-> H+ + HCO3-.

Upon addition of acid;

H+ reacts with HCO3-, forming H2CO3, shifting the equilibrium to the left.

Upon addition of base;

OH- reacts with H2CO3, forming water and HCO3-, shifting the equilibrium to the right.