One reason to care about the cohomological dimension of a group comes from etale cohomology, because etale cohomology over a field is the same thing as Galois cohomology. (Namely, if $k$ is a field, etale sheaves on $\operatorname{Spec} k$ are the same thing as continuous discrete $G$-sets for $G$ the Galois group of the separable closure, and cohomology of an abelian sheaf is (profinite) group cohomology.) As a result, if one wants to prove vanishing theorems in etale cohomology (the most important of which states that for a variety of dimension $n$ over an algebraically closed field, the cohomology groups of any torsion sheaf vanish in degrees >2n) the basic case one is often reduced to is that of a field. It thus becomes necessary to find bounds for the cohomological dimension of Galois groups.
(Henceforth, I am using the definition of cohomological dimension for torsion modules.)
To actually compute this, one can use the following fact: $G$ has cohomological dimension $\le n$ if and only if, for each $p$, there is a $p$-Sylow subgroup $G_p \subset G$ such that $H^{n+1}(G_p, \mathbb{Z}/p) = 0$. The justification is that any finitely generated $p$-torsion $G_p$-module has a finite filtration with quotients isomorphic to $\mathbb{Z}/p$, and after that one can use restriction and inflation to get the result for $G$. In practice one way to show the vanishing of these groups is to use certain exact sequences, for instance
$0 \to \mathbb{Z}/p \to k^{sep* } \to k^{sep *} \to 0$
where the last map is raising to the $p$th power. (When $p$ is the characteristic, this should be replaced by $a \mapsto a^p -a $ and one uses the additive group.) Since there are many theorems on the cohomology of $k^{sep*}$ (keywords: Brauer group, Tate's theorem, Hilbert's theorem 90) and that of $k_{sep}$ (this is actually trivial by the normal basis theorem), one can often use them to get results about cohomological dimension.
A very fun reference for bounding cohomological dimension (but with no mention of etale cohomology) is Serre's book "Galois cohomology."