I think the main case of use for this notation is for matrix groups. As in your example, if $G\subset GL(n,\mathbb R)$ is a closed subgroup, then you can view the inclusion of $G$ as a smooth function $g:G\to M_n(\mathbb R)$. Taking the exterior derivative, you get $dg\in\Omega^1(G,M_n(\mathbb R))$ and your example is exactly this form for $G=GL(n,\mathbb R)$. This is mainly used to express the Maurer-Cartan forms (which encode the trivializations of $TG$ by left or right translations). Since these translations are the restrictions of linear maps, it follows that the left Maurer Cartan form can be written as $g^{-1}dg$ (i.e. to compute its action on a tangent vector, apply $dg$ and then multiply from the left by the inverse matrix of $g$). For the right Maurer Cartan form, one similarly obtians $(dg)g^{-1}$. I think these notations for the Maurer Cartan forms are sometimes also used for general Lie groups, but in my opinion this is rather confusing.