Examples of group-theoretic results more easily obtained through topology or geometry

The Kaplansky Conjecture asserts that the group ring $\Bbb Q G$ contains no non-trivial zero-divisors when $G$ is a torsionfree group.

It is implied by the Atiyah Conjecture, (a version of) which states that for every compact connected CW complex $X$ with $\pi_1(X)=G$, all $\ell^2$-Betti numbers are integers.

The Atiyah Conjecture is now known to be true for a great deal of groups - in particular for groups for which the truth of the Kaplansky Conjecture was unknown before.

There is an exact sequence, due to Hopf: $$\pi_2(X) \to H_2(X) \to H_2(\pi_1(X)) \to 0,$$ where the first map is the Hurewicz map. In some sense, $H_2(\pi_1(X))$ measures how surjective the (2-dimensional) Hurewicz map is. The purpose of this answer is to sell you on this exact sequence. Here's a small result you can prove with it.

Theorem: if $G$ is the fundamental group of a homology sphere, then $H_1(G) = H_2(G) = 0$. In particular, this is true of the binary icosahedral group (a double cover of $A_5$), which is the fundamental group of the Poincare sphere.

Proof: $H_1(G) = G^{ab} = H_1(X) = 0$. That $H_2(G) = 0$ follows immediately from the above exact sequence, because $H_2(\pi_1(X))$ is a quotient of $H_2(X)$.

Two notes.

1) I don't know an easier or different way at all to prove this fact.

2) This property actually characterizes fundamental groups of homology spheres. It is an old theorem of Kervaire that if $G$ is a finitely presented group with trivial $H_1$ and $H_2$, then for fixed $n > 4$, you can find a homology $n$-sphere with fundamental group $G$. $n=3,4$ are (as usual!) something of a mystery.

Here's a completely algebraic question: what do the subgroups of a free product $G_1 \ast G_2 \ast \dots \ast G_n$ look like?

The Kurosh subgroup theorem implies, among other things, that they are all free products of copies of $\mathbb{Z}$ and of subgroups of each of the $G_i$. This generalizes the Nielsen-Schreier theorem, which you get when each of the $G_i$ is $\mathbb{Z}$. The idea of the topological proof is to convert the question into a question about covering spaces of a wedge sum of spaces with fundamental groups $G_1, G_2, \dots G_n$, and it turns out to be possible to describe these covering spaces as generalized graphs in some sense (which can be made precise e.g. using Bass-Serre theory). The copies of $\mathbb{Z}$ show up when these graphs have loops in them.

For example, the subgroups of the modular group $\Gamma \cong PSL_2(\mathbb{Z}) \cong \mathbb{Z}_2 \ast \mathbb{Z}_3$ are all free products of copies of $\mathbb{Z}, \mathbb{Z}_2$, and $\mathbb{Z}_3$. In fact you can even make precise claims about which groups of this form can appear with which indices using the notion of virtual or orbifold Euler characteristic; see, for example, this blog post for some details (and pictures).