I'm looking for a basic motivation for this topic. Meaning, why would one want to study the function field of a curve. The reason I usually give to this question is that one understands a space by understanding the functions that can be defined on it. Also, this object categorically classifies curves up to birational equivalence.
However, to motivate a younger student who doesn't know anything about birational geometry nor indeed has yet developed the intuition about understanding geometry through functions, these reasons can seem arbitrary. Does anyone know of a way or analogy to see why one would come to study function fields? In particular, of curves?