Simple explanation for what a torsor is

Oversimplified & in a nutshell:

1. Recall the slogan

   A $G$-torsor is like the group $G$ that has forgotten its neutral element.

2. Example: An affine space $A$ is a torsor for a vector space $V$.

3. The space of $U(1)$ gauge fields is an affine space, while $\Omega^1(X)$ is a vector space.