I'm not sure if I know a good source, but I can make a remark.

In the category of topological spaces up to homotopy, there is a highly nontrivial geometric invariant that people have taken as interesting: What is the lowest dimension of a topological representative of a given homotopy type? For instance if you have a group $G$, what is the lowest possible dimension of a classifying space $B_G$? The cohomological dimension is a lower bound for this geometric invariant which is more algebraic and sometimes more tractable.

For example, if you have a group $G$, it may not be easy to tell whether it is the fundamental group of a compact, hyperbolic $n$-manifold. (Or a Euclidean manifold or a non-positively curved manifold.) If you can compute its group cohomology, then you'll learn something about it, because there is such an $n$-manifold, then the cohomological dimension of $G$ is exactly $n$.

As another example, if $G$ is a non-trivial finite group, then it is not hard to compute that its cohomology is periodic and thus its cohomological dimension is infinite. Thus, any finite-dimensional CW complex with a non-trivial but finite fundamental group has to have higher homotopy groups.

Maybe a good reference is Brown's GTM book on cohomology of groups.

I'm not sure if I know a good source, but I can make a remark.

In the category of topological spaces up to homotopy, there is a highly nontrivial geometric invariant that people have taken as interesting: What is the lowest dimension of a topological representative of a given homotopy type? For instance if you have a group $G$, what is the lowest possible dimension of a classifying space $B_G$? The cohomological dimension is a lower bound for this geometric invariant which is more algebraic and sometimes more tractable.

For example, if you have a group $G$, it may not be easy to tell whether it is the fundamental group of a compact, hyperbolic $n$-manifold. (Or a Euclidean manifold or a non-positively curved manifold.) If you can compute its group cohomology, then you'll learn something about it, because there is such an $n$-manifold, then the cohomological dimension of $G$ is exactly $n$.

As another example, if $G$ is a non-trivial finite group, then it is not hard to compute that its cohomology is periodic and thus known that its cohomological dimension is infinite. Thus, any finite-dimensional CW complex with a non-trivial but finite fundamental group has to have higher homotopy groups.

Maybe a good reference is Brown's GTM book on cohomology of groups.