countably locally finite covers
It means the latter: the union is a cover (and a refinement of the original) but the locally finite subfamilies that form the union are just known to be locally finite, no more.
Another term for such a family is $\sigma$-locally finite, analogous to terms like $\sigma$-compact. It avoids the suggestion these families are countable or something like that. The sigma prefix means countable union in these terms.
So read it as (countably-(locally finite)) cover.