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I do not have the book with me. The proof is somewhere there as @YCor mentioned. But the statement is not difficult. Assign to each element of a countable group $G$ a natural number $0,1,2,...$ (for every $n$ only finite number of elements $g$ are assigned the same numbers $n(g)=n$, $n(1)=0$ and 1 is the only element with number 0, $n(g^{-1})=n(g)$). Then define a norm $|g|$ of $g$ as the smallest sum $n(g_1)+...+n(g_k)$ such that $g=g_1g_2...g_k$ in $G$. Then define the distance $dist(g,h)=|g^{-1}h|$. It is left invariant and has finite balls.