Understanding proof of uniqueness in theorem on polynomial interpolation

You don't know that it has exactly $n+1$ zeros, just that it has at least that many because you have found them: $x_0, x_1, \ldots, x_n$. But an $n^{\text{th}}$ degree polynomial can only have $n$ zeros, unless it is identically zero...

The phrase "Since the $x_i$ are distinct, $p_n - q_n$ has $n+1$ zeros" is intended to mean that there are at least $n+1$ zeros. It would have been a little better to say "at least," but being absolutely precise can be a bit tedious.