Why are sections of sheaves called sections? Generally, a section is a right inverse. On the other hand, if $F$ is a ($\mathsf{Set}$-valued) sheaf, then the elements of $FU$ are usually also called sections.
Why is this terminology justified and how is it related to right inverses?
Please do not rely on any theory of bundles (I don't know any).
 A: All this is very classic, you can find it in the book of Hartshorne for example. 
Assume we have a space $X$ and an local homeomorphism $\pi : Y \to X$. For every open $U \subset X$ we can look at the continuous maps $s : U \to Y$ such that $\pi \circ s = id_U$. These maps are "sections" in the classical sens, and of course you can take the restriction. Therefore, this space gives you a sheaf $\mathscr S(U) := \{s:U \to Y \mid  \pi \circ s = id_U, \text{s is continous} \}$. The canonical examples are the covering spaces (or vector bundles,etc...).
But in fact every sheaf can be obtained in this way. Now assume you have a sheaf $\mathscr F$ on some space $X$. You can construct a space $Y$, called the étalé space of $\mathscr F$. As a set, $Y = \cup_{p \in X}\mathscr F_x$. 
We obtain a covering map $\pi :Y \to X$. We define the topology on $Y$ in the following way : if $f \in \mathscr F(U)$ then $[U,f]$ be all the germs $s_y$ such that $s_y = f_y$ and $y \in U$. Then, all the $[U,f]$ form a basis for a topology, and $\pi$ is now a local homeomorphism. We just have to see that we recover the original sheaf but the stalk $\mathscr F_x \cong \mathscr S_x$ are naturally isomorphic so every sheaf arise as the sheaf of sections of an étalé space.
So a sheaf could be seen exactly as a map $\pi : Y \to X $ which is a local homeomorphism. In fact, this is the definition taken in the book of Godement, or FAC for example.
