This post assumes knowledge of fiber bundles, the group action functor, groupoids, and basic vector calculus. I am in the process of learning the topics discussed below, and I deeply appreciate constructive feedback.
How does a big space cover a little one?
Given a covering space $E \to B$ we can uniquely lift any path in the base space (once you choose a starting point) to a path in $E$. Conversely, we can create a covering space of $B$ by letting the fiber over $b$ be $F(b)$.