The transformations can be deduced essentially from representations of Lie algebras. For the Lagrangian you have written, we are describing a $U(1)$ gauge theory. Irreducible representations of $U(1)$ are one complex dimensional and are always of the form $e^{i q \alpha}$ where $q\in {\mathbb R}$. By definition, this is a representation and therefore acts on a one complex dimensional vector space as
$$
\phi \to e^{i q \alpha} \phi
$$
We can now apply this principle in physics where everything is a function of space-time, $\phi \to \phi(x)$, $\alpha \to \alpha(x)$. You can check that writing $\phi = \frac{1}{\sqrt{2}} ( \phi^1 + i \phi^2)$ reproduces precisely the transformations you have written.
The transformation of $A_\mu$ can then be deduced from geometric considerations. To do this, let us try and construct a Lagrangian that is invariant under the local gauge transformations
$$
\phi (x) \to e^{i q \alpha(x)} \phi(x)
$$
The first natural thing to do is to try and write down a kinetic term for the scalar. However, the usual one, namely $\partial_\mu \phi \partial^\mu \phi$ no longer works since it transforms weirdly under gauge transformations. The reason of course is that the derivative of a field is defined as
$$
n^\mu \partial_\mu \phi(x) = \lim_{\epsilon \to 0} \frac{ \phi(x+ \epsilon n ) - \phi(x) }{ \epsilon}
$$
Clearly, the issue is that in the derivative, we are taking a difference of fields at different space-time points! Since the gauge transformation is local, it acts differently on the fields at different space-time points. This is basically the issue. To remedy this, we introduce a different field $W(x,y)$ such that $W(x,x) = 1$ and under gauge transformations transforms as
$$
W(x,y) \to e^{i q \alpha(x) } W(x,y) e^{- i q \alpha(y) }
$$
We can define a "new derivative" as
$$
n^\mu D_\mu \phi(x) = \lim_{\epsilon \to 0} \frac{W(x,x+\epsilon n) \phi(x+ \epsilon n ) - \phi(x) }{ \epsilon}
$$
Take the limit described above and you will reproduce precisely the covariant derivative you have in your Lagrangian where
$$
W(x,x+\epsilon n) = 1 - i \epsilon n^\mu A_\mu + {\cal O}(\epsilon^2)
$$
You can also further check that the gauge transformation of $W$ implies the specific gauge transformation of $A_\mu$ that you wrote down.
Thus, all the gauge transformations can be "derived" from arguments like above. Note also that this discussion can easily be generalized to non-abelian gauge theories.
DISCLAIMER: I HAVE NOT KEPT TRACK OF FACTORS OF $e$ AND SIGNS. PLEASE FIX THEM YOURSELF.
