I have been trying to understand Cartan connections based on the Wikipedia article, and I am confused about the following paragraph (here $P$ is a principal $H$-bundle with $H$ a Lie group):
The pair $(\omega, \theta)$ (a principal connection and a solder form: $\hspace{2pt}\omega:TP \to \mathfrak h$, $\theta: TP \to R^n$) defines a 1-form $\eta$ on $P$, with values in the Lie algebra $\mathfrak g$ of the semidirect product $G$ of $H$ with $R^n$, which provides an isomorphism of each tangent space $T_p P$ with $\mathfrak g$. It induces a principal connection $\alpha$ on the associated principal $G$-bundle $P\times_{H}G$. This is a Cartan connection.
(Note: the above is part of the article that serves to motivate Cartan connections; it is not the full, general definition). What does the $H$ subscript in $P\times_{H}G$ in the last line mean? My guess from the context is that $P\times_{H}G$ means to swap out the fibers of $P$ with $G/H$. But if this were correct, then the action of $G$ on the resulting fibers would not be free, and hence $P\times_{H}G$ would not be a principal $G$-bundle, whereas the last line refers to it as such.
The bundle denoted by $P\times_H G$ is the associated (fiber-) bundle to the principal $H$-bundle $P$ corresponding to the left action of $H$ on $G$ defined by multiplication from the left. This is the natural way to extend the structure group of a principal bundle. A Cartan connection on $P$ can indeed be viewed as a principal connection on this extended bundle which satisfies a non-degeneracy condition. In my opinion, this point of view looses some information, however, unless you remember that there is a also the specified $H$-subbundle $P\subset P\times_H G$.
EDIT: There are various ways to see the relation between a Cartan connection and the homogeneous space $G/H$. Most simply, the defining properties of a Cartan connections on a principal $H$-bundle $P\to M$ are obtained by weakening the properties of the Maurer-Cartan form on the principal $H$-bundle $G\to G/H$ in such a way that they make sense on a general $H$-bundle.
The description you are looking for probably is what is often called "Cartan's space $\mathcal S$" in the literature. This is the associated bundle $P\times_H (G/H)$ where $H$ acts on $G/H$ by the restriction of the $G$-action. The latter fact implies that one can also view thist as $(P\times_H G)\times_G (G/H)$ so it can be viewed as an associated bundle to the extended principal $G$-bundle. Since the extended principal $G$-bundle carries a principal connection, $\mathcal S$ inherits a (fiber-bundle-)connection, which gives a notion of parallel transport along curves, etc., thus connecting the copies of $G/H$ attached to different points of $M$. But this is a rather complicated geometric object (it encodes higher order data and is non-linear at the same time), so it is a bit complicated to deal with.