It is said in this blog that:
The reason this turns out to be useful is that many examples in algebraic/arithmetic geometry require you to look no further than understanding modules over Dedekind domains.
Could anyone provide me some examples illustrating this?
There are many possible answers. Since you posted this in algebraic geometry, I will give you a geometric answer.
Suppose that you wanted to understand a regular affine curve $C=\text{Spec}(A)$. A natural thing that you might want to do is study line bundles over $C$ since this tells you quite a bit of geometric information about $C$. The standard way to study such line bundles is to look at the Picard group of $C$ which is the group of isomorphism classes of line bundles (under tensor product). So, how can we compute the class group, $\text{Pic}(C)$?
One possible way is through the more general notion of $K_0(R)$ (the $0^{\text{th}}$ K-group of $R$) for a ring $R$. This is merely the group of isomorphism classes of finitely projective $R$-modules. For example, as an exercise, convince yourself that $K_0(k)=\mathbb{Z}$ for $k$ a field (or, significantly harder, for $k$ a local ring). While this seems like a pretty useless definition, it's use can be seen through the following neat theorem:
So, for example, if $R=\mathcal{O}_K$, for some number field $K$, then we know that $\text{Cl}(\mathcal{O}_K)=\text{Pic}(\text{Spec}(\mathcal{O}_K)$ is finite and so
$$\text{Cl}(\mathcal{O}_K)=\text{Pic}(\text{Spec}(\mathcal{O}_K))=K_0(\mathcal{O}_K)_\text{tors}$$
So, if you understood all of the modules over a Dedekind domain (even just the projective ones!) you'd understand completely the set of line bundles over your domain. Or, if you are more arithmetically minded, you can think about just the above centered equation, and realize that you can understand the class group of a number ring purely by studying modules over it.
If you want to learn more about algebraic K-theory, I would highly recommend Rosenburg's book on the subject.