Atiyah K-theory

Let $p : E \to B$ be a (real for example) vector bundle. It's locally trivial, meaning that if you take a small open set $U \subset B$, then you can find an isomorphism $\varphi : p^{-1}(U) \cong U \times \mathbb{R}^n$, compatible with $p$ (meaning that $\varphi(\xi) = (p(\xi), \text{something})$).

Now a section $s : B \to E$ of $p$ is a map such that $p \circ s = id_B$. So if you restrict to the small open set $U$, then $s(b) \in p^{-1}(\{b\}) \subset p^{-1}(U)$, therefore $\varphi(s(b)) = (p(s(b)), \text{something}) = (b, \text{something})$. Call the "something" $\sigma(b)$ (it depends on $b$), then you get a map $\sigma : U \to \mathbb{R}^n$ (determined by $s$). This is what is meant by "locally a section is given by a vector valued function on the base space": the vector valued function is $\sigma$.