The way we define an associated bundle to a principle bundle is by constructing the trivial bundle over the principle bundle with the desired fibre and then quotienting out by the structure group (encoding a representation in the equivalance classes). This way we get a bundle whose fibers are objects that transform in a certain way under the action of the structure group.
In other words, if $P$ is a principle bundle, $F$ is the fibre on which the structure group has a representation, the associated bundle is $P \times F / \sim_G$.
Now I understand in the gauge theories of physics like Yang-Mills or GR, we have a principle bundle and consider gauge transformations of it. These are automorphisms of the principle bundle, elements of the group $\mathrm{Aut(P)}$.
Equivalently (this makes sense locally, but why also globally?), they are changes in the local trivialization of the principle bundle generated by different (but intersecting) patches in its base space, say $M$. If we have a connection $A$ on $P$ and two trivializing neighborhoods, the gauge fields (pullbacks of connection over two different trivializing sections) $\sigma^{*}_1A$ and $\sigma^{*}_2 A$ are related by the usual gauge transformation rule which schematically looks like $$\sigma^{*}_2 A = g^{-1}(\sigma^*_1 A) g+g^{-1}dg$$
My question: matter fields, which are realized as sections of associated bundles are made for the purpose of transforming a certain way under elements of the structure group. But why does this sort of principle bundle automorphism business above cause transformation in the associated bundles. Naively, I see that the shuffling around of structure group elements would mean for an element of a fibre in the associated bundle to represent the same geometric thing, the representative of the equivalence class defining it would change under the representation of $G$. But this isn't precise.