The meaning of the WP quote is that a divisible subgroup of an abelian group is always a direct summand. On the other hand, any divisible abelian group is a direct sum of copies of $\mathbb{Q}$ and $\mathbb{Q}_p/\mathbb{Z}_p$, thus unlike general infinite abelian groups, they can be completely classified. See the books of Kurosh and Kargapolov–Merzlyakov on group theory for the proofs. As Steve D mentioned in the comments, there is a straightforward generalization to modules over PIDs.
Here are two related "concrete applications outside of algebra".
1 Pontryagin duality for locally compact abelian groups, $G\mapsto\text{Hom}(G,T),$ where $T=\mathbb{R}/\mathbb{Z}$ is the circle group and Hom denotes continuous homomorphisms.
2 Tate duality for Galois modules in algebraic number theory, $A\mapsto\text{Hom}(A,\mu)$, where $\mu$ is the group of roots of unity ($\mu\simeq\mathbb{Q}/\mathbb{Z}$ as abelian groups, but the Galois module structure is different).
In both cases, divisibility of the target is needed to assure that the natural map into the double dual is injective.