That calculation is doubly useful because it also helps explain why the pH of the 0.10 M acetic acid/sodium acetate mixture seems to be the same as the 1.0 M mixture.

Clearly, adding the "common ion", in this case acetate from the sodium acetate, can have an effect on a weak acid (and presumably also on a weak base). The common ion is, of course, just the conjugate base of the weak acid (or the conjugate acid of the weak base). The word weak is important here. When solid NaCl is added to 0.10 M HCl there is no apparent change in pH based on our experiment. But adding solid NaCH₃COO to 0.10 M CH₃COOH raises the pH enough to make the indicator change color. Why?

The ability to manipulate pH in this way is more than a curiosity. When weak electrolytes and their conjugates are mixed a chemical system is established which is able to regulate pH to a certain extent.