In my work I come upon divisible abelian groups all the time, especially those of the form $\mathbb Q_p/\mathbb Z_p$, or direct sums of such groups. One frequently uses the fact that they are injective (to conclude that certain sequences of Hom groups, which a priori would only be left exact, are in fact exact on the right as well), and other properties as well; for example, groups of type mentioned above, related to $\mathbb Q_p/\mathbb Z_p$, have non-zero $p$-adic Tate modules, which can often be useful.

This last property is related to the role that divisible groups play in duality theory (in the sense of Pontrjagin duality), and this is another reason that divisible groups are important.

I think I probably speak for a lot of algebraic number theorists when I say that a lot of what we do is a kind of applied algebra. We don't care about algebraic properties of structures for their own sake, but in our work we encounter lots of different groups, rings, and modules, and we like to understand their properties as well as we can, so that we can use them to gain control of the number theoretic computations that we are trying to make. Divisibility of an abelian group is one such useful property, which we are trained to recognize and exploit.