Gauge symmetries, as the note says, are redundancies in our description of nature. For example, a photon has two physical degrees of freedom (the two polarizations). However, we choose to describe a photon using a 1-form field $A_\mu$ which has 4 degrees of freedom. The two extra degrees of freedom here are related to gauge symmetries. From here on, there are two important questions to be answered:
Why must gauge symmetries be preserved in the quantum theory?
One of the things that should be preserved in going from the classical to a quantum theory is the degrees of freedom of the theory. (The way the degrees of freedom behave can change, but not their number). Taking the example of the photon, the total unphysical degrees of freedom remains unchanged upon quantization (the quantum and classical photon is described by the same $A_\mu$). Thus, in order to maintain the right number of physical degrees of freedom (2 for the photon) gauge invariance must be preserved upon quantization.
If gauge symmetries are redundancies, why introduce them in the first place?
The important point here is the requirement that our theory be Lorentz invariant. Now, there are two ways to be certain that our theory (or more precisely, the $S$-matrix) is Lorentz invariant
Make the action manifestly Lorentz invariant. This implies describing the theory in terms of Lorentz covariant objects such as scalars $\phi$, 1-forms $A_\mu$ or spinors $\psi$ (and other representations of the Lorentz group.)
Don't make the action manifestly Lorentz invariant, i.e. formulate it in terms of fields that are not representations of the Lorentz group, but maintain overall Lorentz invariance by carefully putting together the fields in a Lorentz invariant way.
You may immediately see the advantages of the first technique. Lorentz invariance is manifest, and as long as the indices match up nicely, we never have to worry about getting funny Lorentz non-invariant answers. With the second technique, one has to check for Lorentz invariance at every step of the calculation.
The disadvantage of the first method is the following: Every physical object must be embedded in representations of the Lorentz group. Thus, if we want to describe a spin-1 photon, it must be embedded in the spin-1 representation of the Lorentz group, $A_\mu$. This leads us to a necessary introduction of gauge invariance. (since the 4 dof of $A_\mu$ has to be reduced to the 2 of the photon)
So as a summary, while gauge invariance does create some inconveniences, it allows to circumvent the even more inconvenient formulation of the theory in a Lorentz non-manifest way.
PS - In recent developments, people have tried embedding the dof of the photon into the spinor, which continues to allow for manifest Lorentz invariance, but also circumvents the problem of gauge invariances. This is called the spinor-helicity formalism.