I wouldn't start with QFT, but rather read some classical electromagnetics. The idea there is that Maxwells equations are invariant under a certain type of symmetry, which is called a gauge symmetry since it is a transformation in field space and has no spacetime components.
This symmetry stems from the fact that electromagnetism is a relativistic theory, so gauge invariance is an inherently relativistic effect.
If you want to understand this again in more detail, it is (somewhat) back to QFT, consider e.g. the first book by Weinberg: there you see that it makes sense for your particles to lie in a certain irreducible representation of the Lorentz group (there is relativity again), and for massless particles (connection to electrodynamics here) there are some Lorentz transformations that correspond to gauge symmetry (so called Wigner translations).
In other gauge theories (SU(3) for example), things are more complicated because your fields have a more complex inner structure, but the basic idea is the same.
So my suggestion: first classical electromagnetism, then, when you understood its emergence there, take on the book by Weinberg, after this you will have underdstood it.
Good luck!
