The question is equivalent to asking for the ratio of the measures $\mu(a \mathcal{O})/\mu(\mathcal{O})$, where $\mathcal{O}$ is the valuation ring and $a \in K^{\times}$. But in a $p$-adic field, whenever you have one ball centered at zero contained in another ball centered at $0$, the larger ball is simply a disjoint union of translates of the smaller ball. Since the Haar measure is translation invariant, the ratio of the volumes just comes out to be the number of translates. If $a \in \mathcal{O}$, then the number of translates of $a \mathcal{O}$ in $\mathcal{O}$ is nothing else than the cardinality of the quotient ring $\mathcal{O}/a \mathcal{O}$.
I leave it to you to work out the problem according to the notation in the book (especially since I think your comment trying to clarify the notation may be mistaken). But for instance in $\mathbb{Q}_p$, the ratio of the volume of $\mathcal{O}$ to $p \mathcal{O}$ is equal to $p$, because $\# \mathcal{O}/p\mathcal{O} = \# \mathbb{Z} / p \mathbb{Z} = p$. In general this ratio will be multiplicative in $a$, not additive (so it should be a $K$-adic norm, not a $K$-adic valuation).