Motivation for construction of associated fiber bundle from a principal bundle

To answer your specific question, you get a principal $H$-bundle, often denoted $P \times^G H$. The transition maps are clearly just given by applying $\phi$ to those of $P$.

I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.

This construction reverses the construction of the frame bundle from a vector bundle (e.g., the tangent bundle). The idea is that each point $f \in F_p$ in the frame bundle of a vector bundle $E$ is, by definition a basis of $E_p$. This therefore defines a natural map of $F \times \mathbb{R}^k \rightarrow E$, where $k$ is the rank of $E$. The inverse image of any $e_p \in E$ is naturally isomorphic to $G$, so the quotient $(F\times \mathbb{R}^k)/G$ is a vector bundle isomorphic to $E$.

This is just an attempt to elaborate a bit on Ben McKay's answer beyond the confines of a mere comment.

Principal $G$-bundles $P(M,G)$ over $M$ can be understood as a sort of "universal generator" of transition cocycles for its associated $G$-bundles over $M$. More precisely, the transition cocycles associated to a $G$-bundle atlas for $P(M,G)$ are transition cocycles for the associated $G$-bundle with any given typical fiber $F$, up to the choice of $G$-action on $F$. The quotient $(P\times F)/G$ is meant to achieve precisely that. This is especially clear in the case of frame bundles and vector bundles, as depicted in Deane Yang's answer.

Since the family of transition cocycles encode all information on the topological deviations of $P\times_G F$ from the trivial bundle $M\times F$, this means that all nontrivial topological information of $P\times_G F$ in this sense is already encoded in $P(M,G)$.