I am not clear about what you find confusing. The answer to the first question seems simply to be that the expression $g^{-1}dg$ is meaningless on an abstract Lie group. To be more precise, the expression $g$ is subjected to the exterior derivative $d$ by first being interpreted as a function on the group $g \colon G \to GL(n)$, i.e. the inclusion function, if $G\subseteq GL(n)$. But on an abstract Lie group, there is no such inclusion function. So we need to find a property of $g^{-1}dg$ that holds on matrix Lie groups $G\subseteq GL(n)$, and which can generalize to all Lie groups.
I think Deane answered the other questions. For more detail, the definition of a wedge product of 1-forms $\alpha,\beta$ valued in an arbitrary algebra $A$ (not necessarily a Lie algebra) with multiplication $x\in A, y \in A \mapsto m(x,y)\in A$ is $m(\alpha,\beta)(v,w)=m(\alpha(v),\beta(w))-m(\alpha(w),\beta(v))$. I don't think there is any more reasonable definition. When applied to the Lie algebra of matrices under bracket, this yields $[\alpha,\beta]=2\alpha\wedge\beta$, if you expand it out. Hence the factor of 2 to relate Lie bracket to matrix multiplication.
We need the Lie bracket because, as Deane says, there is no other obvious product in the Lie algebra.
Edit: maybe I didn't give enough intuition. The intuition behind left translation is clear. Left translation identifies all tangent spaces of any Lie group. But that makes the definition of Maurer-Cartan form obvious: you give Maurer and Cartan your velocity as a vector in a tangent space of $G$, and they hand you back the corresponding velocity vector at the identity element, so you can compare velocities at different points. But that is just the definition of Maurer-Cartan form, and is clearly just $g^{-1} dg$ on a matrix Lie group: the $g^{-1}$ in $g^{-1}dg$ is the left translation back to the identity.
We derive the equation for $d\omega$ as in lots of books: $d\omega(X,Y)=L_X(\omega(Y))-L_Y(\omega(X))-\omega([X,Y])$, applied to left invariant vector fields, using the left invariance of $\omega$ to kill the first two terms.
I insist that the superior approach to the study of moving frames is to avoid ever taking a local section $f$. Instead, to each immersed submanifold $\iota \colon M \to X=G/H$, we associate the pullback bundle $\iota^*G$, and the pullback Maurer-Cartan form. We think of this pullback bundle as the bundle of adapted frames (or coframes). This generalizes easily to immersed submanifolds in Cartan geometries and to foliations of Cartan geometries. Whatever you are reading about the moving frame method, throw your source away and work it out from the beginning using bundles and reduction of bundles.