The way a global field has always been described to me is a field that is either a finite extension of $\mathbb{Q}$, or a field of rational functions on a projective curve over a field of characteristic $p$ (i.e., finite extensions of $\mathbb{F}_q(t)$).
This definition never made much sense to me. Sure, we use it all the time and can come up with some pretty cool results using it, but intuitively it never made sense why those two classes of fields are related. On a high level, what is the connection between these two types of fields?