In the The principal connection for a connection form section of Wikipdia Connection form article, first there is a local trivialization $\{U_k,φ_k\}$ with structure group $G$ of vector bundle $(E,π,M)$, choosen local $G$-frames $\{e_k\}$ which are related by $e_V=e_U\cdot h_{UV}$ where $h_{UV}$ are some $G$-valued functions defined on $U\cap V$.
Then construct the associated principal $G$-bundle $F_GE=\bigsqcup_U U\times G/\sim$ where the equivalence relation $\sim$ is defined by:
$$((x,g_U)\in U\times G)\sim((x,g_V)\in V\times G) \iff g_U=h_{UV}g_V$$
(note: in the article it is $g_U=h_{UV}^{-1} g_V$ which is against the transition function tradition and also prevent to form the global form later so should be a bug)
Then obviously here $h_{UV}$ is not just "some" $G$-valued functions, but is indeed the transiton function of the given local trivialization.
The problem is, commonly the transformation relations between the arbitrarily choosen local $G$-frames $\{e_k\}$ have nothing to do with the transition functions $h_{UV}$, since the former is choosen freely on the given local trivialization $\{U_k,φ_k\}$ but the latter is already decided by the choosen local trivialization.
Then if the article is interpretated literally, first the local $G$-frames $\{e_k\}$ are choosen freely and they are related by "some" $G$-valued functions $h_{UV}$, then contruct the principal $G$-bundle with these functions as "transition functions", i.e. $g_U=h_{UV}g_V$. Note that in this interpretation the transition function of the local trivialization $\{U_k,φ_k\}$ will not be the same as $h_{UV}$, which is very not reasonable: you should use the same local trivialization and transition function to construct an associated bundle. That is why this literal interpretation should be wrong.
The non-literal interpretation is: here indeed a very special type of local $G$-frames are choosen, which transfoms exactly by the transition functions $h_{UV}$ of the choosen local trivialization. Is that true and how to prove that such special local $G$-frames always exist?