Like any theory, Galois theory matured with time. It was understood that the theory can be expressed more clearly in terms of fields and many applications outside of the original confines the theory was designed for emerged. This is also part of a general trend in modern mathematics, namely abstracting and axiomatizing.
This is not unique to Galois theory. Linear algebra was originally the study of systems of linear equations, but today we concentrate on the concept of an abstract vector space. Analysis used to be very much concerned with the study of certain metric spaces, e.g., spaces of functions, but the concept of an abstract metric space is, today, central. Measure theory was designed to produce a better behaved integral than the Riemann integral, but today the concept of an abstract measure space is fundamental.
Basically any new theory is designed to answer some question and thus is fraught with specific details that can, and should, be abstracted away so that we can see the forest and not just trees.