U(1) Gauge Symmetry: What Informs Its Choice?

[dfttheory]

So, I have a basic/general question here. I understand that, for example, the QED Langrangian has U(1) gauge symmetry. I also understand that this means (when you have written the Lagrangian with the covariant derivative) that if you transform the wavefunction ($\psi \rightarrow e^{i \theta (x)} \psi$) and the covariant derivative, this Lagrangian remains invariant.

What I don't understand is this: what does it mean for the wavefunction to have this local symmetry? How do we know that electrons / photons are described by this theory? What principal of nature says that the wavefunction has this symmetry?

I know this is three questions, but I am just trying to get a sense of what informs the choice of symmetry in these theories before I continue transforming and writing gauge invariant theories.

Thank you for any attention you may pay this question!

 

[dfttheory]

I just realized that the discussions related to this may contain some of what I am looking for. I would be interested in any additional discussion about this. Thanks!