Throughout we work in the smooth setting.
Let $\pi:E\to M$ be a rank $k$ real vector bundle and $F(E)\to M$ denote the corresponding frame bundle.
I am trying to understand the following statement, which is taken from this wikipedia article, under the heading "Relation to Principal and Ehresmann Connections."
The sections of $E$ are in one to one correspondence with the equivariant maps $F(E)\to \mathbf R^k$. (This can be seen by considering the pullback of $E$ over $F(E)\to M$, which is a trivial bundle isomorphic to $F(E)\times \mathbf R^k$).
I don't see how the correspondence comes about, and especially how the triviality of the pullback bundle gives it. What I can see is that if we have a section $\sigma$ of $E$, then we can define a map $F(E)\to \mathbf R^k$ which sends $(p, T)\in FE_p$ to $T^{-1}(\sigma(p))$. But I am unable to get a map in the reverse direction.
Can somebody please help, especially elucidating as to how the triviality of the pullback comes to our rescue.