Here is a physical interpretation of the associated bundle.
In gauge field theory, the symmetry group $G$ of a principal bundle $G\rightarrow P\rightarrow M$ is interpreted as the thing that dictates the possible interactions of elementary particles, in the following way. Given a representation of $G$ on a vector space $V$, one may construct the associated vector bundle as the orbit space $E=(P\times V)/G$ with respect to the action you defined in your question. Now $E$ comes equipped with a natural structure of a vector bundle. The sections of this vector bundle are interpreted as the states of the particles that your model describes (e.g. the scalar Higgs boson). When describing the force carriers, you utilize the notion of a connection on your principal bundle. These are 1-forms on $P$; pulled back (via a local section of $P$), these are interpreted as the states of the force-carriers that your model describes (e.g. the weak W- and Z-bosons).
This is the mathematical setting for describing the physical interactions between force carriers and scalar bosons (both are bosons, the former are the so-called gauge bosons): equipping your principal bundle with a connection, you end up with a natural construction of a covariant derivative on $E$, which describes the coupling of elementary particles. You even end up with some Feynman diagrams.
Visualising interactions between gauge and scalar bosons with Feynman diagrams.
In this way, we may describe how force carriers obtain mass, via the so-called mechanism of "spontaneous symmetry breaking". Finally, interactions of bosons and fermions are described by using additionaly the spin structure of your base space $M$.