As mention in David Tong's Lecture note:

Gauge symmetry is, in many ways, an odd foundation on which to build our best theories of physics. It is not a property of Nature, but rather a property of how we choose to describe Nature. Gauge symmetry is, at heart, a redundancy in our description of the world. Yet it is a redundancy that has enormous utility, and brings a subtlety and richness to those theories that enjoy it.

So how does Spontaneous Symmetry Breaking of local gauge symmetries that leads to massive gauge bosons make sense when in fact it is not a real symmetry at all?

As mention in David Tong's Lecture note:

Gauge symmetry is, in many ways, an odd foundation on which to build our best theories of physics. It is not a property of Nature, but rather a property of how we choose to describe Nature. Gauge symmetry is, at heart, a redundancy in our description of the world. Yet it is a redundancy that has enormous utility, and brings a subtlety and richness to those theories that enjoy it.

So how does Spontaneous Symmetry Breaking of local gauge symmetries that leads to massive gauge bosons make sense when in fact it is not a real symmetry at all?