https://www.math.cornell.edu/~hatcher/VBKT/VB.pdf
Reading through Hatcher's proof of the the induced exact sequence of $\widetilde{K}$ groups, I've run into a few issues.
I'm unsure of how there is an induced projection $E/h \to X/A$, I tried forming some type of commutative diagram, but was unsuccessful.
He also says that "A trivialisation of $E$ over $A$ determines the sections $s_i : A \to E$ which form a basis in each finer over $A$." I'm not sure how such a section is determined, and why we are guaranteed that it forms a basis in each fiber.
The last thing I don't understand is "Via a local trivialisation, each section $s_i$ can be regarded as a map from $A \cap U_j$ to a single fiber".
Regarding the induced projection. If I am not mistaken, the equivalence relation is defined to be $h^{-1}(x,v)\sim h^{-1}(y,v)$, and we have maps $E\overset{p}{\to} X\overset{q}{\to} X/A$. To obtain a map $E/h \to X/A$ we need to show that $q\left(p\left(h^{-1}\left(x,v\right)\right)\right)=q\left(p\left(h^{-1}\left(y,v\right)\right)\right)$, but $p\left(h^{-1}\left(x,v\right)\right)=x\in A$, and similarly for $y$, and indeed $q$ maps take them to the same point $*\in X/A$.
To simplify notations, let's denote $B=p^{-1}(A)\subseteq E$, and for $a\in A$ denote $F_a=p^{-1}(a)$ (the fiber over $a$). A trivialization is a map $h:B \to A\times \mathbb{C}^n$, which satisfies some conditions. For each $e_i\in \mathbb{C}^n$ define $s_i: A \to E$ by $s_i(a)=h^{-1}(a,e_i)$ (which is indeed a section of $p$).
One of the conditions $h$ must satisfy is that for each $a\in A$, $h\mid_{F_a}:F_a\to \{a\}\times \mathbb{C}^n\cong\mathbb{C}^n$ is an isomorphism (of vector spaces), so since the $e_i$-s are a basis for $\mathbb{C}^n$, it follows that the $s_i(a)$-s are basis for $F_a$.
The trivialization $h$ allows you to project everything to $A \times \mathbb{C}^n$, and by further projection, to any of its constituents. So you can this to obtain from $s_i\mid_{A\cap U_j}: A\cap U_j \to B$, a new map $A\cap U_j \to \mathbb{C}$.
Now you can apply the Tietze extension theorem, to extend if to a map $U_j \to \mathbb{C}$.
Using the trivialization yet again, you get the extend map $s_{ij}: U_j \to E$.