On the Wolfram Mathworld article for Commutator Subgroup, it states that the first homology group is the abelianization, $$H_{1}(G) = G \big/ [G,G]$$ which totally blows my mind because I've only seen the commutator subgroup in the context of Lie algebra representation theory, and I've only seen the first homology group in the context of simplicial homology. This leads me to the following question,
Can you give an explanation (rigorous, or intuitive) for why the first homology group is the abelianization? Furthermore, if you happen to know of how simplicial homology may be related to Lie algebra representation theory (or know of a reference regarding it), please tell!
$H_1(G)$ is the first homology group of the corresponding Eilenberg-Maclane space $K(G,1)=X$ (which is by definition the space X such that $\pi _1(X)=G$ and all other homotopy groups of X are trivial). Denote $X=K(G,1)$ Then your question reduces to the following
$H_1(X)=\pi_ 1(X)/[\pi _1(X),\pi _1(X)].$
This is a standard algebraic topology fact which you can find in ant standard Algebraic topology textbook (e.r. Algebraic topology by Hatcher Theoem 2A.1)