I think the following is used in classifying $F$-torsors.
Let's take $0\to F\to E\stackrel{\varphi}{\to} L\to 0$ to be an exact sequence of vector bundles, all over a scheme $S$, where $L=\mathbb{A}^1_S$ is the trivial line bundle over $S$. If $$\sigma\colon S\to L$$ is defined by $s\mapsto 1$ for all $s\in S$ is the constant section sending everything to $1$, why does it follow that $\varphi^{-1}(\sigma(S))$ is an $F$-torsor? Thanks.
I think this is just a rephrasing of the fact that affine subspaces of a vector space $V$ are torsors over a linear subspaces. Here by an affine subspace I mean a subspace of $V$ of the form $W = \{w:Aw = c\}$ where $A : V \to k$ is a linear function and $c \in k$. This is a torsor over the linear subspace $K = \{v: Av = 0\} = \ker A$ where $K$ acts freely and transitively by translation since for $w \in W$ and $v \in K$, $A(w + v) = Aw + Av = c + 0$ so $w + v \in W$.
So your statement is the globalization of this linear algebra fact. $\varphi^{-1}(\sigma(S))$ is the affine space bundle which in each fiber over $s \in S$ consists of the affine subspace $\{\varphi_s(x) = 1\}$ which is a torsor over $\ker{\varphi_s} = F_s$.