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 $G$ and transiton functions $\{h_{ij}\}$ of vector bundle $(E,π,M)$, choose local $G$-frames $\{e_k\}$ which are related by $e_V=e_U\cdot h_{UV}$.
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 construct a form on each $U\times G$:
$$ω_{U(x,g)}=Ad_{g^{-1}}π_1^*ω(e_U)+π_2^*ω_{\mathfrak g}$$
where $π_1$ and $π_2$ are projections from $U\times G$ to $U$ and $G$ respectively , $ω_{\mathfrak g}$ is the Maurer-Cartan form, $ω(e_U)$ is the local connection form on $U$ by local section $e_U$.
Then the purpose is to prove that for any point $x\in U\cap V$ and $(x,g_U)\sim (x,g_V)$ there is $ω_{U(x,g_U)}=ω_{V(x,g_V)}$ or equivalently:
$$Ad_{g_U^{-1}}π_1^*ω(e_U)=Ad_{g_V^{-1}}π_1^*ω(e_V)\tag{1}\label{1}$$
Since $e_V=e_U\cdot h_{UV}$ there is $ω(e_V)=Ad_{h_{UV}^{-1}}ω(e_U)+h_{UV}^*ω_{\mathfrak g}$ (from the Structure groups section of that article), put it into the right of $\eqref{1}$:
\begin{align} Ad_{g_V^{-1}}π_1^*ω(e_V) &=Ad_{g_V^{-1}}π_1^*(Ad_{h_{UV}^{-1}}ω(e_U)+h_{UV}^*ω_{\mathfrak g})\\ &=Ad_{g_V^{-1}}π_1^*Ad_{h_{UV}^{-1}}ω(e_U)+Ad_{g_V^{-1}}π_1^*h_{UV}^*ω_{\mathfrak g}\\ &=Ad_{g_V^{-1}}Ad_{h_{UV}^{-1}}π_1^*ω(e_U)+Ad_{g_V^{-1}}π_1^*h_{UV}^*ω_{\mathfrak g}\\ &=Ad_{g_V^{-1}h_{UV}^{-1}}π_1^*ω(e_U)+Ad_{g_V^{-1}}π_1^*h_{UV}^*ω_{\mathfrak g}\\ &=Ad_{g_U^{-1}}π_1^*ω(e_U)+Ad_{g_V^{-1}}π_1^*h_{UV}^*ω_{\mathfrak g}\\ \end{align}
The first part of this is the same as the left in $\eqref{1}$, but there is an extra term $Ad_{g_V^{-1}}π_1^*h_{UV}^*ω_{\mathfrak g}$ that is obviously not zero. What's the problem?