$U(1)$ Local Gauge Invariance: What do $q$ and $alpha(x)$ mean?

A generic element $U$ of a gauge group $G$ can be written in term of the exponential of the generators $T_a$ of the group:

begin{equation} U=e^{ialpha_a T_{a}} end{equation}

where a sum over $a$ is intended. In gauge theory we promote $alpha$ to $alpha(x)$. In our case the group is $U(1)$ and we have only one generator $Q$, which is a number:

begin{equation} U=e^{ialpha(x)Q} end{equation}

So $alpha(x)$ is not the fine structure constant, it is just an arbitrary real function of $x$. The generator $Q$ is interpreted as the (total) electric charge.

Gauge invariance then becomes equivalent to the conservation of charge. In fact, if $E(phi(x)_i)$ is some expression containing product of fields, each of those fields, under a $U(1)$ transformation, will transform as

begin{equation} phi(x)_ito e^{ialpha(x)Q_i}phi(x), hspace{5mm} phi^{*}(x)_ito e^{-ialpha(x)Q_i}phi^{*}(x) end{equation}

Hence

begin{equation} E(phi(x)_i) to e^{i(Q_1+Q_2+...)}E(phi(x)_i) end{equation}

For the expression to be invariant, we have then to require that

begin{equation} sum Q_i =0 end{equation}

which is the statement of conservation of charge.

About the last statement of the page: it turns out that when you evaluate loop diagrams (representing some scattering or annihilation process), the integrals representing them are infinite. You could circumnavigate this problem by absorbing the infinities into counterterms added to the Lagrangian. This has the effect of modifying the couplings. In particular, the value of the electric charge gets some modifications. This is a pure quantum effect, it does not appear in tree level processes (which can be interpreted classically).