Why is the action of $SL(2, R)$ on holomorphic quadratic forms involve a square root? Following this, given a quadratic differential $q$, it can be identified with the pair of real 1-forms $(\Im(q^{1/2}), \Re(q^{1/2}))$. Given a matrix $A \in SL(2, R)$, it acts on this pair by left multiplication, and we define $A \cdot q$ to be the quadratic differential identified with this pair.
My question is: Where does the $1/2$ power of $q$ in the aforementioned pair of 1-forms come from?
 A: Note that the text assumes $q$ is a quadratic differential, not an abelian one.
I would like to explain the differences, motivate quadratic differentials a bit
and show how they fit in the context of dynamical systems.  In the meantime,
I will answer your question, if you have not already seen the point of the square root.
Let's start with abelian differentials, which I will take to be holomorphic one-forms
on a closed Riemann surface $S$ of genus $g$.  What do we do with one-forms, what role 
do they play?  
They are objects you can integrate along paths on $S$ without having to
worry about coordinate charts: they are global objects, locally given by expressions
of the form $f(z)dz$ where $f$ is holomorphic in whichever local chart $z$ you pick.
Furthermore, if $f(z)$ has a zero at some point in the chart, near that point the differential
takes the form $z^kdz$, where $k$ is the order of the zero.  
Back to integration: if you have an abelian differential $\omega$ on $S$, let's pick a point
$p$ on $S$ where $\omega$ is not zero and a tiny simply connected open set $U$ around $p$ 
where $\omega$ continues being non-zero.  Form the function $z: S \to \mathbb{C}$ defined as
$$z(a) = \int_p^a \omega.$$  I did not specify the path, because I did not have to: by 
Cauchy's theorem and the holomorphicity of $\omega$, any two paths from $p$ to $a$ 
in the simply connected $U$ will give me the same result.
Look now: $z$ is a local chart from $U$ to $\mathbb{C}$ and $dz =\omega$ in $U$ by definition.
That is, our abelian differential provided us with a nice local chart.
But this chart is more than nice.  If we consider another point $q$, $W\ni q$ open and
simply connected, $\omega = dw$ constructed as above in $W$ and $U\cap W\neq \emptyset$ say connected and simply connected, 
one should have a transition function between charts.
Can you see what this transition function is?  On $U$, $$z(a) = \int_p^a \omega; $$ on $W$
 $$w(b) = \int_p^a \omega;$$ therefore, for each $r$ in the intersection of $U$ and $W$,
 $$w(r)-z(r) = \int_p^q\omega =C $$ a constant!  
Upshot: an abelian differential provides an atlas of charts with transition functions being just
translations.  But where do dynamics come in?
In dynamical systems, we care about real curves on surfaces.  Along these curves flow important
dynamical systems, so we want to understand the topology and geometry of the curves on our surface, at least
up to a local perturbation that remains small if you flow along $S$.
Returning to $U$, $z$ and $\omega=dz$, note that the image in the complex plane of our chart can be 
split into its real and imaginary parts.  Let's appropriately label them $x$ and $y$: $x= \Re(z)$, $y=\Im(z)$,
so $dx=\Re(dz)$ and $dy=\Im(dz)$.  Looking infinitesimally around the image of $p$ under $z$ which we can take 
to be zero, the level sets $x=c$ and $y = c$ provide a pair of transverse foliations of the image which are
the simplest thing imaginable: horizontal lines and vertical lines respectively.
Now here comes the crucial bit: since the transition functions between $z$ and $w$ are translations,
the set of horizontal lines is preserved in the intersection, and so is the set of vertical lines:
the foliations are not just local things, they are global!  The big assumption is 
that $\omega$ does not have any zeros around there, but I will neglect this for now.  The crucial point
is that the pair of transverse foliations in the charts is compatible with transition functions
and thus patches up to a pair of global transverse foliations on $S$!  
This is the fundamental data for a translation surface, a smooth surface $S$ with a pair of foliations
that determine a consistent vertical and a consistent horizontal direction, along which 
dynamical systems related to the topology of $S$ can flow (such as a geodesic flow in the prescribed directions).  (You need to allow a finite number of exceptional marked points for most $S$, but this comes after you have understood the difference between abelian and quadratic differentials).
But there is a nagging problem in this discussion: if I give you a vertical foliation in $\mathbb{C}$, just
vertical straight lines, and a crossing horizontal one, just horizontal straight lines, sure translations
preserve these two sets, but they are not the only kind of transformation of $\mathbb{C}$ that does:
rotations of angle $\pi$ also preserve these two sets ($\pi/2$ does not, it switches the sets).  Our abelian
differentials are unable to accommodate that symmetry of the foliations.  
For some $S$, this is not a big problem, depending on the topology.  But for most $S$, this missing symmetry
is really losing us information.  This is where quadratic differentials come to play.
If I rotate by 180 degrees my picture, algebraically I am performing the substitution $z\mapsto -z$.  Accordingly,
$dz\mapsto -dz$ and indeed while I have a symmetry of my foliations, I am changing differentials.  The brutish 
way to mend this is just square the negative away: $(dz)^2 = (-dz)^2$ even when the local chart is rotated.  Quadratic
differentials are the global refinement of this process.  Locally they look just like $(dz)^2$ away from zeros.  Some
quadratic differentials can even be written as $q = (\omega)^2$ globally, where $\omega$ is an abelian differential.
Not all can.
This way, with quadratic differentials we accommodate all obvious symmetries of the two local foliations, and patch
them together in a global manner.  Dynamically speaking, the difference between the abelian and quadratic case is
that: if within a chart, you are flowing on say a straight horizontal line, then when you exit the chart, next time
you come back in the abelian case you will be flowing perhaps on a different horizontal line in the same direction,
whereas in the quadratic case you may also be switching direction. 
There are ways to avoid quadratic differentials using appropriately chosen covers of $S$, but I will not enter that
discussion.  I also have not mentioned the crucial issue of zero loci, but since your question displayed a confusion
about abelian versus quadratic differentials, these topics are beyond it.
To conclude: the textbook you are reading takes a quadratic differential, locally $(dz)^2$, takes its local square root,
locally $\sqrt{dz^2} = dz$, takes real and imaginary parts, locally $dx$ and $dy$ defining infinitesimal generators
of the two foliations, and applies a matrix in $\textrm{SL}(2,\mathbb{R})$ to distort these generators and therefore
the resulting pair of foliations.  This provides a possibly different differential and therefore a possibly different
flat structure on $S$.  As you will read later on, the moduli space of these flat structures is related to the moduli
space of quadratic differentials and the dynamics on $S$ is crucially related to the dynamics of a certain flow
on that moduli space and its simply connected cover, the Teichmuller space.
