Let $\pi:P\rightarrow M$ be a Principal $G$-bundle and let $P\times_G E$ be its associated bundle via the representation $\rho:G\rightarrow GL(E)$.
We know that we can identify the set of sections of the associated vector bundle $\Gamma(P\times_G E)$ with the set of functions ${C^\infty(P,E)}^G:= \lbrace f:P \rightarrow E: f(pg)=\rho(g^{-1})f(p) \rbrace$.
In page 19 in the book Heat kernels and Dirac operators, the authors say:
Let $P$ be a principal bundle with structure group $G$. There is a representation of the sections of an associated vector bundle $P \times_G E$ as functions on the corresponding principal bundle that is extremely useful in doing calculations in a coordinate free way.
I'm interested in understanding why and how is this identification useful ?
I believe if you keep reading the book the statement you are asking about will justify itself; for instance on p.24 the isomorphism $C^\infty(P; E)^G\cong C^\infty(P\otimes_\rho E\to M)$ is used to define a covariant derivative on $P\otimes_\rho E\to M$ from a connection $1$-form on $P$ since one can think of $C^\infty(P; E)^G$ as $E$-valued $G$-equivariant differential $C^\infty$ $0$-forms on $P$.
Probably this perspective is not developed in the book in question but the very same idea ("sections of associated bundles are given by graphs of functions") is also very useful in measure rigidity/ ergodic theory (e.g. in Margulis' Measurable Factor Theorem).