I agree that the claimed formula does not seem to make sense without using some identification of $\psi_j$ and $\psi_i$ as matrices. If one chooses some fixed local trivialization of $E$, then one might of course do so, and then what is written in the proof after equation (4.4) seems to be a derivation of how the two gluing conditions are equivalent (and thus the first one is independent of the choice of fixed trivialization).

To prove the proposition, I would suggest instead to start with your equation for the gluing, compose with $\psi_j^{-1}$ from the left and $\psi_j$ from the right, which yields
$$
\psi_j^{-1} \psi_{ij}^{-1} d\psi_{ij} \psi_j = \psi_j^{-1} A_j \psi_j - \psi_i^{-1} A_i \psi_i
$$
and then conclude the proof as in the last paragraph of Proposition 4.2.19.