Why does gauge invariance have physical consequences? My understanding is that gauge invariance occurs when the description of a physical field as a mathematical field (i.e., function whose domain is space-time) contains a redundancy: there are infinitely many possible values of the mathematical field that represent the same physical field. Furthermore, there must be "local" freedom in the sense that the set of mathematical field configurations corresponding to a physical field configuration is itself parametrized by another mathematical field.
Presented this way, gauge invariance seems to be a mere artifact of the method of mathematical description chosen. However, it seems that we can deduce actual physical laws from it, e.g., Maxwell's equations from the electromagnetic gauge invariance. How can a mere mathematical artifact have observable physical laws as consequences?
 A: When we start to describe a physical system in a particular mathematical framework, our choice of framework is motivated by physical observations. It turns out that to describe the quantum theory of a massless spin 1 particle such as the photon, the quantized version of a classical field theory with local symmetry is highly effective. Our physical motivation: why is the photon massless? 
Gauge invariance in a quantum field theory has two physical consequences (that we note here, there are other consequences as well). The first: the Ward-Takahashi identities, which are nontrivial operator identities arising from the demand that gauge symmetry hold as an exact symmetry, order by order in perturbation theory. The second: spontaneous breaking of such symmetries is a good way to describe massive spin 1 particles within the framework of a renormalizable quantum field theory that has a classical Lagrangian description. As you mentioned, it is an artifact of description: other descriptions of the same physical system may see a different local symmetry in its version of the classical Lagrangian. However, the effects in the full quantum theory, i.e., the operator identities, will still be physical observables, i.e., the spectrum will still have a massless degree of freedom. Contrast this with massless $\phi^4$ theory, where there is no gauge symmetry, and radiative corrections generate a mass and break the classical scaling symmetry. It is not sufficient to write down a classical Lagrangian without a mass term, to ensure that the quantum theory describes a massless field. This is what gauge symmetry does, it helps us implement a physically observed operator relation in the language we are most familiar with: perturbative QFTs. 
