On the page of commutator subgroup Wikipedia says that "the larger the commutator subgroup is, the "less abelian" the group is."
I know that for every group $G$ and $ N\trianglelefteq G$ the quotient $G/N$ is Abelian if and only if $[G,G] \le N$.
But what does "the larger the commutator subgroup is, the "less abelian" the group is" mean?
It's a somewhat vague comment, which nevertheless makes some sense. Since the commutator is, as you noticed, the minimum you need to "mod out by" in order to get an abelian group, its size does indeed correspond to how far the group is from being abelian.