Transition Functions
Definition. Where two trivializations overlap, $\Phi_\alpha \circ \Phi_\beta^{-1}(x, f) = (x,\, t_{\alpha\beta}(x) \cdot f)$ defines $t_{\alpha\beta}: U_\alpha \cap U_\beta \to G$, valued in the structure group. They satisfy the cocycle condition $t_{\alpha\beta}\,t_{\beta\gamma} = t_{\alpha\gamma}$ on triple overlaps.
Intuition. The gluing instructions. A bundle is its transition data: given a cover of $M$ and functions $t_{\alpha\beta}$ satisfying the cocycle condition, you can reconstruct $E$ by gluing patches $U_\alpha \times F$. The bundle is trivial iff $t_{\alpha\beta} = g_\alpha g_\beta^{-1}$ for some functions $g_\alpha: U_\alpha \to G$ (the twist can be "absorbed").
Examples. Möbius: two patches, $t = \pm 1 \in \mathbb{Z}_2$. Dirac Monopole: north/south hemispheres of $S^2$, $t_{NS} = e^{in\phi}$ on the equator — the winding number $n$ is the monopole charge.
Physics anchor. For $U(1)$: $t_{\alpha\beta} = e^{i\theta_{\alpha\beta}(x)}$, and the local potentials glue by $A_\beta = A_\alpha + d\theta_{\alpha\beta}$ — Landau's gauge shift (§18), now revealed as patch-to-patch bookkeeping, mandatory whenever the bundle is nontrivial.
Related: Local Trivialization · Fiber Bundle · Connection 1-Form (gluing law) · Dirac Monopole
Sources: Sternberg Ch. VII.