Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be an open covering for $M$ that yields local trivialisations with transition maps $$ \varphi_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow GL(n,\mathbb{R}). $$ Then $D$ defines a $T^*M$-valued matrix $A_\alpha$ on $U_\alpha$. Let a section $s$ be given locally on $U_\alpha$ by $s_\alpha=s^i_\alpha\mu_i$, where $\{\mu_1,...,\mu_n\}$ is a frame for $E_{|_U}=\pi^{-1}(U)$.
Question I: Why does it hold that $$ s_\beta=\varphi_{\beta\alpha}s_\alpha\qquad\text{on $U_\alpha\cap U_\beta$}? $$
Question II: Why does it follow that $$ \varphi_{\beta\alpha}(d+A_\alpha)s_\alpha=(d+A_\beta)s_\beta\qquad\text{on $U_\alpha\cap U_\beta$}? $$ (He does give an "indication" of how this holds, but I don't see what he means.)
Question III: How do we then conclude that $$ A_\alpha=\varphi_{\beta\alpha}^{-1}d\varphi_{\beta\alpha}+\varphi_{\beta\alpha}^{-1}A_\beta\varphi_{\beta\alpha}? $$
Remark: Jost states this in each of his books on geometry, but I have never been able to find an elaboration. I would also be grateful for some other references that explain in more detail what is going on here.
The following is taken more or less directly from section 4.1 of Jost's Riemannian Geometry and Geometric Analysis. (Where he uses $\mu$, I will use $s$ to match the OP's notation.)
I think you may be getting confused when you try to introduce the frame $\{ \mu_i \}$. I don't think there's any need to mention frames here.
Consider a section $s$ of $E$. On each $U_\alpha \subset M$ over which $E$ is trivial, we can represent $s$ by a local section $s_\alpha$. This means that $s_\alpha$ is a map $U_\alpha \to \mathbb{R}^n$, i.e., a vector-valued function on $U_\alpha$. $\varphi_{\beta \alpha}$ is a map $U_\alpha \cap U_\beta \to Gl(n, \mathbb{R})$, i.e., for each point $p \in U_\alpha \cap U_\beta$, $\varphi_{ \beta \alpha}(p)$ is an invertible linear map $\mathbb{R}^n \to \mathbb{R}^n$. $s_\alpha$ and $s_\beta$ are related, for $p \in U_\alpha \cap U_\beta$, by $s_\beta (p) = [\varphi_{\beta \alpha}(p)] ( s_\alpha(p))$ (to answer your Question I, this is essentially just the definition of the transition map $\varphi_{\beta \alpha}$). (I will drop the reference to the point $p$ from now on.)
Now, on to the connection $D$. As Jost discusses and as you mention, $D$ defines locally on $U_{\alpha}$ a matrix $A_\alpha$ with one-form entries. Again, we use the local trivialization over $U_\alpha$ to view our section $s$ locally as a map $s_\alpha: U_\alpha \to \mathbb{R}^n$. $D s$ is a section of $E \otimes T^\ast M$, meaning locally $Ds$ is a sum of terms of the form $\sigma \otimes \omega$, where $\sigma$ is a section of $E$ and $\omega$ is a one-form. If we write the "$E$-piece" of $Ds$ locally in terms of the local trivialization of $E$, we can view $Ds$ locally as a vector with one-form entries that I'll call $(Ds)_\alpha$ (my notation, not Jost's). As Jost discusses, $(Ds)_\alpha = ds_\alpha + A_\alpha s_\alpha$, where $d$ is the usual exterior derivative acting componentwise on the entries of the vector-valued function $s_\alpha$, and $A_\alpha$ acts on $s_\alpha$ by matrix multiplication to give a vector with one-form entries.
Now to your Question II. On $U_\alpha \cap U_\beta$, we can write $Ds$ as a vector with one-form entries in two different ways corresponding to the two local trivializations: $(Ds)_\alpha$ and $(Ds)_\beta$. These two ways should be compatible in the sense that applying $\varphi_{\beta \alpha}$ to $(Ds)_\alpha$ should give us $(Ds)_\beta$, and this is where the equation you wrote comes from: \begin{align*} \varphi_{\beta \alpha} ((Ds)_\alpha) &= (Ds)_\beta, \text{ i.e.,} \\ \varphi_{\beta \alpha} ((d+ A_\alpha) s_\alpha) &= (d+ A_\beta) s_\beta \end{align*}
Finally, we substitute the fact that $s_\beta = \varphi_{\beta \alpha} s_\alpha$ into the above equation to solve for $A_\alpha$ in terms of $A_\beta$. This answers your Question III: \begin{align*} \varphi_{\beta \alpha} ((d+ A_\alpha) s_\alpha) &= (d+ A_\beta) (\varphi_{\beta \alpha} s_\alpha) \\ \varphi_{\beta \alpha} (d s_\alpha) + \varphi_{\beta \alpha} (A_\alpha s_\alpha) &= (d\varphi_{\beta \alpha})s_\alpha + \varphi_{\beta \alpha} (ds_\alpha) + A_\beta \varphi_{\beta \alpha} s_\alpha \text{ (using Leibniz rule for $d$)}\\ \varphi_{\beta \alpha} (A_\alpha s_\alpha) &= (d\varphi_{\beta \alpha})s_\alpha + A_\beta \varphi_{\beta \alpha} s_\alpha \\ A_\alpha s_\alpha &= \varphi_{\beta \alpha}^{-1} (d\varphi_{\beta \alpha} + A_\beta \varphi_{\beta \alpha} ) s_\alpha \\ \end{align*} This holds for every $s_\alpha$ (the section $s$ was arbitrary), so we have the following equality of matrices with one-form entries on $U_\alpha \cap U_\beta$: $$ A_\alpha = \varphi_{\beta \alpha}^{-1} (d\varphi_{\beta \alpha} + A_\beta \varphi_{\beta \alpha} ) $$