The Corollary 3.39 on page 250 says:
If $M$ is a closed connected orientable $n$-manifold, then an element $\alpha \in H^k(M;\mathbb{Z})$ generates an infinite cyclic summand of $H^k(M;\mathbb{Z})$ iff there exists an element $\beta \in H^{n-k}(M;\mathbb{Z})$ such that $\alpha \smile \beta$ is a generator of $H^n(M;\mathbb{Z}) \approx \mathbb{Z}$.
My question regards what he means by "infinite cyclic summand". If we are assuming finitely generated homology groups then this statement makes sense to me using the classification of finitely generated abelian groups. However, no assumption has been stated anywhere in the preceding text. What am I missing?
Every compact manifold has finitely generated homology groups, as Hatcher mentions on page 231. You can find a proof in the appendix of the book (Corollaries A.8 and A.9).