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Right now Wikipedia defines it as:

Let $E\rightarrow M$ be a smooth vector bundle over a differentiable manifold $M$. Let $\Gamma(E)$ be the space of all smooth sections. A connection on $E$ is a $\mathbb{R}$-linear map $\nabla:\Gamma(E)\rightarrow\Gamma(E\otimes T^*M)$ such that $$\nabla(\sigma f)=\nabla(\sigma)f+\sigma \otimes df$$ for all smooth functions $f$ on $M$ and all sections $\sigma$.

What I don't understand is how $\sigma f$ is defined. If $\sigma\in\Gamma(E)$ and $f\in C^\infty(M)$, why would $\sigma f$ be in $\Gamma(E)$? For that matter, I also don't understand how $\nabla(\sigma)f$ is defined.

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    A section is a map : $s : M \to E$ with $s(m) \in E_m$. So it makes sense to multiply the section by a scalar. And a function $f : M \to \mathbb R$ is precisely the data of a scalar (moving smoothly) so it makes sense to take the section $f \cdot s(m) := f(m)s(m)$ if $s$ is a section. – 2017-01-16

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A section $\sigma \in \Gamma(E)$ gives you a smooth choice of elements $\sigma(p) \in E_p$ for each $p \in M$ where $E_p$ is the fiber of $E$ over $M$ which is a vector space. Given a smooth function $f$, you can multiply each $\sigma(p)$ by the scalar $f(p)$ and obtain another smooth section $f\sigma$ in $\Gamma(E)$. By definition, $\nabla \sigma$ is a section of the vector bundle $E \otimes T^{*}M$ so it can be multiplied by a smooth function as above. The only reason this multiplication is written on the right (as $(\nabla \sigma) \cdot f$) and not on the left (as $f \cdot (\nabla \sigma)$) is to make the defining property of $\nabla$ look like a Liebnitz rule considering the type of $\nabla$.