This is from Hatcher: Let $(E,B,p)$ is a vector bundle.
If one has $n$-linearly independent sections, the map $h:B \times \mathbb{R}^n \to E$ given by $h(b,t_1,···,t_n)= \sum_i t_i s_i(b)$ is a linear isomorphism in each fiber, and is continuous since its composition with a local trivialization $p^{-1}(U) \to U \times \mathbb{R}^n$ is continuous.
I've had no trouble up til now. This part just seems vague to me. How is the composition continuous?
With regards to your specific query, the local trivialisations commute with the fibrewise linear structure on $E$. So if $\varphi:p^{-1}(U)\xrightarrow{\cong}U\times \mathbb{R}^n$ is such a local trivialisation then for each $b\in U$ we have
$$\varphi\left(\sum_{i=1}^n t_i\cdot s_i(b)\right)=\sum_{i=1}^n t_i\cdot \varphi (s_i(b))$$
where the right-hand side is interpreted as the sum in $\mathbb{R}^n\cong\{b\}\times\mathbb{R}^n$. The point is that the sections $s_i$ are continuous with respect to the variable $b\in U$ so if for each $i=1,\dots,n$ we let $\hat s_i:U\rightarrow \mathbb{R}^n$ be the composite
$$\hat s_i:U\xrightarrow{s_i} E_U\xrightarrow{\varphi}U\times \mathbb{R}^n\xrightarrow{pr_2} \mathbb{R}^n$$
then we get a family of continuous $\mathbb{R}^n$-valued maps on $U$, and our first equation tells us that the composite $\varphi\circ h|_{U\times\mathbb{R}^n}:U\times\mathbb{R}^n\rightarrow U\times \mathbb{R}^n$ is equal to the composite
$$U\times \mathbb{R}^n\xrightarrow{\Delta\times 1}U^{n+1}\times\mathbb{R}^n\xrightarrow{shuf}U\times (\mathbb{R}\times U)^n\xrightarrow{1\times\prod(1\times\hat s_i)}U\times (\mathbb{R}\times \mathbb{R}^n)^n\xrightarrow{1\times m^n}U\times (\mathbb{R}^n)^n\xrightarrow{1\times\oplus}U\times\mathbb{R}^n$$
where $\Delta:U\rightarrow U^{n+1}$ is the $(n+1)$-fold diagonal, the second map $shuf$ shuffles the coordinates appropriately, $m:\mathbb{R}\times\mathbb{R}^n\rightarrow \mathbb{R}^n$ is scalar multiplication and $\oplus:\mathbb{R}^n\times\dots\times\mathbb{R^n}\rightarrow\mathbb{R}^n$ $(x_1,\dots,x_n)\mapsto x_1+\dots+x_n$ is the iterated vector addition in $\mathbb{R}^n$.
It should be clear from this presentation that $\varphi\circ h$ is continuous (and even smooth if you work in the $C^\infty$ category).