The physical reason to introduce gauge symmetry is a combination of two facts:

1. We are working in the context of a theory that obeys special relativity (Lorentz invariance) and quantum mechanics
2. We observe massless spin-1 particles and so want to describe them in our formalism,
3. We want our formalism to encode local interactions (since, in special relativity, non-local interactions are also not causal).

Point 3 leads us to wanting to use a vector field $A_\mu(x)$ to describe the spin-1 field, since then it is easy to write a Lagrangian with local interactions by multiplying fields at the same spacetime point.

However, in order to describe a masslesss spin-1 particle, there should be 2 propagating degrees of freedom, yet $A_\mu$ has 4 components. Gauge invariance (or gauge redundancy) is introduced to project out the two "unphysical" components of $A_\mu$ in a Lorentz invariant way.

Once we have committed to introducing a gauge symmetry to describe our spin-1 particle, then all its interactions should also obey the gauge symmetry -- otherwise the unphysical components of $A_\mu$ will couple to physical particles. "Gauging a local interaction" is a method of generating consistent couplings of a spin-1 field to other matter fields in a way that respects the gauge symmetry.

The physical reason to introduce gauge symmetry is a combination of two facts:

1. We are working in the context of a theory that obeys special relativity (Lorentz invariance) and quantum mechanics
2. We observe massless spin-1 particles and so want to describe them in our formalism,
3. We want our formalism to encode local interactions (since, in special relativity, non-local interactions are also not causal).

Point 3 leads us to wanting to use a vector field $A_\mu(x)$ to describe the spin-1 field, since then it is easy to write a Lagrangian with local interactions by multiplying fields at the same spacetime point.

However, in order to describe a masslesss spin-1 particle, there should be 2 propagating degrees of freedom, yet $A_\mu$ has 4 components. Gauge invariance (or gauge redundancy) is introduced to project out the two "unphysical" components of $A_\mu$ in a Lorentz invariant way.

Once we have committed to introducing a gauge symmetry to describe our spin-1 particle, then all its interactions should also obey the gauge symmetry -- otherwise the unphysical components of $A_\mu$ will couple to physical particles. "Gauging a global symmetry" is a method of generating consistent couplings of a spin-1 field to other matter fields in a way that respects the gauge symmetry.