Section 13.1.2 of Ravi Vakil: "Fix a rank $n$ vector bundle $\pi:V\rightarrow M$. The sheaf of sections $F$ of $V$ is an $\mathcal O_M$-module - given any open set $U$, we can multiply a section over $U$ by a function on $U$ and get another section."
I don't understand what's going on here. A section of $F$ over $U$ is a function $s:U\rightarrow M$ with $\pi s=\iota:U\subset M$. What is meant by a "function on $U$ and how does multiplying by one yield another set theoretic section of $\pi$?
In a vector bundle $\pi:V\to M$, if you take any 'vector' $v\in V$, then this vector is 'over' a point $p:=\pi(v)$, i.e. $v$ is in the fibre $\pi^{-1}(p)$, which is assumed to be a vector space on its own, so that $\lambda\cdot v$ makes sense for any $\lambda\in\Bbb R$, and is in the same fibre.
So, if we have a section $s:U\to V$ with $U\subset M$, and a function $f:U\to\Bbb R$, then $x\mapsto f(x)\cdot s(x)$ makes sense and is also a section, as $$\pi(f(x)\cdot s(x))=\pi(s(x)) = x$$ by the above argument.