The following is from Vakil's notes.
Given a locally free sheaf $\mathcal{F}$ with rank $n$, and a trivializing open neighborhood of $\mathcal{F}$ (an open cover $\{U_i\}$ such that over each $U_i$, $\mathcal{F}|_{U_i} \cong \mathcal{O}^{\oplus n}_{U_i}$ as $\mathcal{O}$-modules) we have transition functions $g_{ij} :U_i \cap U_j \to \text{GL}(n,\mathcal{O}(U_i \cap U_j))$.
Can someone elaborate how these transition functions arise? I understand that we have two different isomorphisms $\varphi_i : \mathcal{F}|_{U_i\cap U_j} \to \mathcal{O}^{\oplus n}_{U_i \cap U_j}$ and $\varphi_j:\mathcal{F}|_{U_i\cap U_j} \to \mathcal{O}^{\oplus n}_{U_i \cap U_j}$ and we can consider the composition $$\varphi_i \circ \varphi^{-1}_j : \mathcal{O}^{\oplus n}_{U_i \cap U_j} \to \mathcal{O}^{\oplus n}_{U_i \cap U_j}$$ but this is a morphism of sheaves and I think that these $g_{ij}$ are maps from $U_i \cap U_j$ so we need to consider $\varphi_i \circ \varphi^{-1}_j$ at the level of sections or something? Any help would be much appreciated.
For our purpose, the indexed open cover doesn't matter so, for now, let's just work on an open subset $U \subset X$.
Then, $GL_n(\mathcal{O}(U))$ is an affine scheme, isomorphic to $\operatorname{Spec} A$ where $$A = \mathcal{O}(U)[x_{11}, x_{12} \dots, x_{nn}]_{\operatorname{det}(x_{ij})}.$$ As such, to give a map $U \to GL_n(\mathcal{O}(U))$ it is equivalent to give a map $A \to \mathcal{O}(U)$.
If $\mathcal{O}^{\oplus n}_{U} \to \mathcal{O}^{\oplus n}_{U}$ is an isomorphism it then gives an isomorphism of $\mathcal{O}(U)$-modules $\psi: \mathcal{O}(U)^{\oplus n} \to \mathcal{O}(U)^{\oplus n}$. Being a map of free modules, it admits a matrix form $\psi = (\psi_{ij})$, with $\psi_{ij} \in \mathcal{O}(U)$, so that $\det(\psi_{ij})$ is a unit in $\mathcal{O}(U)$. By the universal property of localization, the map sending $x_{ij} \mapsto \psi_{ij}$ induces a ring homomorphism $A \to \mathcal{O}(U)$, as required.