Add to your theory/model. In order to make the symmetry work, you need something to "fix" the discrepancy you would otherwise have in the equations. So you add this field (variable) to make the whole thing consistent. In the end, you see you needed this "gauge field" (for historical reasons these are called "gauge symmetries") which is electromagnetism.

Look at the first formula in https://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathem...

If A and B are zero then F is zero and you can forget the second part of the right hand part of the equation. Also, you must replace D with ∂.

Now you have equation of electrons and positrons that move in a universe that has no electromagnetism. The important part is that in that equation, the only variable is ψ(t,x,y,z) that appears twice, the rest of the things written there are just derivatives, constants or indices.

Now you can turn on B(t,x,y,z) that represents the external field, so the electrons and positions move in a more interesting patterns, but they don't "see" each other. Again, the only variable is ψ(t,x,y,z).

Now you do the trick with a local U(1) symmetry, and the only way to do the trick is to add a new variable A(t,x,y,z). But you must use the complete equation because as explained in Wikipedia F(t,x,y,z) is calculated using the derivatives of A(t,x,y,z). So now you have two variables ψ(t,x,y,z) and A(t,x,y,z).

So you "add a new field A" to the list of variables that appear in the right hand of the equation, or to be more precise, you get a new equation that has one additional variable A.