 Solution

Fix a positive rational radius $r$. Distinct disjoint circles of radius $r$ have centers more than $2r$ apart, so their open interior disks are pairwise disjoint. Each such disk contains a point of the countable dense set ${\mathbb{Q}}^2$. Fix an enumeration of ${\mathbb{Q}}^2$ and assign to each disk the first rational point it contains. Disjointness makes this assignment injective, so there are only countably many circles of radius $r$.

The positive rational numbers are countable. Taking the countable union of countable sets over all rational radii proves that the whole collection is countable.