I am having troubles in understanding the modern definition of moving frames method.

Classically, the idea of moving frames is "to express the variation in terms of the moving frame itiself". This is clear if we have a matrix Lie group $G$, like $O(n), GL(n)$...

But if $G$ is not a matrix group, we cannot express the variation in terms of the moving frame. At this point are introduced Lie algebras, Maurer-Cartan forms and all things that I understand but I cannot figure out where they come from.

If $G$ is a matrix Lie group, the Maurer-Cartan form is the usual that I know and so is the Maurer-Cartan equation and that's ok. My question is: why is this the correct generalization of moving frame method to "abstract" Lie group? What's the intuition behind this? Who does this generalization?