Isomorphic to a free abelian group means that it behaves like Zⁿ for some n,that is the set of n-uples of integers with operations defined pointwise.

A simple example is Z², this is the set of all pairs of integers (a,b) where addition is defined as (a,b)+(c,d)=(a+c,b+d).

A more interesting example is the set of pairs (a,b√2) where a and b are integers and addition is defined as (a,b√2)+(c,d√2)=(a+c,(b+d)√2).

If you draw those in R² they look like repeating parallelograms (squares actually for the first), hence lattices