The lecture notes I am reading state that given a principal G-bundle $P\overset{\pi}{\rightarrow} M$, if we are given two patches on the base manifold $U_\alpha$ and $U_\beta$ we have two distinct trivialising maps acting on the fibres over the points in the overlap, $U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}$: $$\psi_\alpha:\pi^{-1}(U_{\alpha\beta})\rightarrow U_{\alpha\beta}\times G, \tag{1}$$ and $$\psi_\beta:\pi^{-1}(U_{\alpha\beta})\rightarrow U_{\alpha\beta}\times G. \tag{2}$$ What is strange to me is that, while these trivialising maps of course map to the same point $m\in U_{\alpha\beta}$, the point $g\in G$ is not necessarily the same. In other words $\psi_\alpha(p)=(m,g_\alpha)$ and $\psi_\beta(p)=(m,g_\beta)$ such that $g_\alpha\neq g_\beta$, it is strange to me that locally, the point $p\in P$ can be locally associated with a different pair $(m,g)$ depending on which patch containing $m$ we are considering on the base manifold.
Disclaimer: I'm a physicist not a mathematician.