Gauge invariance is simply a redundancy in the description of a physical system.  I.e. we can choose from an infinite number of vector potentials in E&M.

For example, an infinite number of vector potentials can describe electromagnetism by the transformation below

$$A(x) \to A_\mu(x) + \partial_\mu \alpha(x)$$

Choosing a specific gauge (gauge fixing) can make solving a physical problem much easier than it would be if you did not fix a gauge.

Normally one chooses the Coulomb gauge: $\nabla \cdot A = 0$.

It should be stressed that gauge invariance is NOT a symmetry of nature and you cannot measure anything associated with it.

Gauge invariance is most useful in quantum field theory and is crucial in proving renormalizability.