In the following proposition
Let $s,t$ be $C^{\infty}$ sections of a $C^{\infty}$ vector bundle $\pi : E \to M$ and let $f$ be a $C^{\infty}$ real valued function on $M$ Then
i) the sum $s + t : M \to E$ defined by $$ (s + t)(p) = s(p) + t(p) \in E_p, \;\; p \in M $$ is a $C^{\infty}$ section of $E$.
ii) the product $fs : M \to E$ defined by $$ (fs)(p) = f(p)s(p) \in E_p, \;\; p \in M$ $$ is a $C^{\infty}$ section of $E$.
I'm having a hard time understanding how this sum and product are actually defined. I know the vector bundle definition involve the vector space structure. However this is through the locally trivialzing map $\pi$ or the fiber preserving map $\phi$. In the definition both $s$ and $t$ are maps from $M \to E$ and I really struggle to understand how the mentioned operations are actually defined.
Can you clarify?
My insight
I'll give you my interpretation. $s,t$ are sections therefore $s(p),t(p)$ belong to the fiber at $p$, name this $E_p$, by definition of tangent bundle, since $E_p = \pi^{-1}(p)$ it has a vector space structure of dimension $r$. It seems apparently that because of this, this is well defined I suppose (can you correct me?). However if I want to actually compute the sum do I need to use the fiber preserving map $\phi$? this indeed would map $E_p$ to $\left\{ p \right\} \times \mathbb{R}^r$ which is a product bundle, can I use the vector space structure of the product bundle somehow?
Given two maps $s,t:M\rightarrow E$, we attempt to construct a new map $M\rightarrow E$.
Let $m\in M$. Then, $s(m)$ is an element of $E$ and $t(m)$ is an element of $E$.
So, the question now is, given two elements of a manifold $E$, how do you produce another element of the manifold $E$. There seem to be no obvious choice.
But, these $s(m),t(m)$ are in a subset of $E$, namely the fibre of $m$ in $E$, denoted by $E_m$ and defined as $$E_m=\{a\in E:\pi(a)=m\}.$$
By definition, this set $E_m$ has structure of a vector space. So, given two elements of $E_m$, one can consider the sum, sum of two vectors in a vector space, to give an element in $E_m$. So, for $m\in M$, we have $$s(m)+t(m)\in E_m.$$
The map $M\rightarrow E$ defined as $m\mapsto s(m)+t(m)$ for $m\in M$, is denoted by $s+t$, called the sum of $s$ and $t$. One can check that this map$s+t:M\rightarrow E$ is a smooth section of $\pi:E\rightarrow M$.
If this is clear, there should be no confusion regarding product of a section by a real valued smooth map.
Let $\Gamma(E)$ denote the set of sections of $\pi;E\rightarrow M$. We are trying to make this set $\Gamma(E)$ into a $C^{\infty}(M)$-module. For this, we need a notion of sum of two elements of $\Gamma(E)$. Additionally, we need product of an element of $C^{\infty}(M)$ with an element of $\Gamma(E)$. Is it clear?