Does it or does it not depend on $x$ alone?
When your text states that $\dot\gamma_0$ 'only depends on the distance $x$ from the edge of the plate', that implies 'from all the variables describing the fluid domain' (that is $x$, $y$ and $z$). This quantity is only defined on the plate, this is a boundary term as opposed to, say, velocity components $u$, $v$ which are defined at any point in the fluid volume. Of course, the explicit expression for $\dot\gamma_0$ could contain other quantities characterizing the problem as a whole: fluid density, viscosity etc. That is why the $x$ variable is separated by semicolon from the rest: it is the only spatial variable here.
I think I found two dimensionless quantities: $x^* = x\dfrac{U\rho}{\eta}$ and $\eta^* = \eta\dfrac{1}{xU\rho}$. When I looked for these quantities I ignored $\dot\gamma_0$ because I don't understand how that function is defined.
Your two dimensionless quantities $x^{*}$, $\eta^{*}$ are inverse of one another, so they are not independent. The five dimensionful quantities have dimensions in units of $M$, $L$ and $T$, so to construct the second independent dimensionless quantity (the first could be either of $x^{*}$, $\eta^{*}$) you must include $\dot\gamma_0$ ($5 = 3 + 2 $).
Even if you do not understand definition of $\dot\gamma_0$ you could find its dimension: it is a derivative of velocity component w.r.t coordinate so its dimension is
$$ [\dot\gamma_0]=\left[\frac{\partial u}{\partial z}\right]=\frac{[u]}{[z]}=\frac{L T^{-1}}{L} =T^{-1}.$$
Incidentally, the (b) part of your question gives a hint on how to construct that second quantity: rewrite the equation
$$ \dot{\gamma}_0 = \dfrac{U^2\rho}{\eta}f\bigg(\dfrac{Ux\rho}{\eta}\bigg) \tag{*}$$
so that the right hand side contains only $f(x^{*})$. Since it is a dimensionless quantity the left hand side would be dimensionless and would be the required second quantity (let us call it $\Gamma$).
Edit: Since it seems to be not obvious, the answer for the second dimensionless quantity:
$$ \Gamma = \frac{\dot\gamma_0 \eta}{U^2 \rho} .$$
Now you have conjectured functional dependence:
$$ \dot{\gamma}_0 =\dot{\gamma}_0(x;\eta,U,\rho),$$
a dimensionless variable $x^{*}$ from arguments of r.h.s and another one, $\Gamma$ from $\dot\gamma_0$, $U$, $\rho$ and $\eta$. So this allows us to apply Buckingham $\pi$ theorem by stating that $\Gamma$ must be a function of $x^{*}$:
$$ \Gamma = f(x^{*}).$$
After simple transformation you would obtain the statement of your (b) part.
Since the $y$-coordinate is irrelevant, we shall omit it. In front of the plate edge, where $x\leq 0$, the flow is uniform with velocity $U$ in the positive $x$-direction. The fluid mass density $\rho$ with $[\rho] = ML^{-3}$ and viscosity $\eta$ with $[\eta] = ML^{-1}T^{-1}$. Close to the plate the friction between plate and flow decelerates the flow. This friction causes the flow velocity near the plate to drop. For increasing $x$, the boundary condition will approach the no-slip condition with the velocity at the plate exactly vanishing. For large $x$ values, the velocity profile will become independent of $x$ and approach a stationary profile, independent of $x$ and thus $t$. We want to know how fast this convergence takes place as a function of time $t$ and thus of position $x$. In the figure above the velocity profiles above the plate are sketched for different $x$-positions. Let the velocity of the flow be denoted by $(u,w)$ with $u$ the velocity in the $x$-direction and $w$ the velocity in the normal $z$-direction. The so-called shear rate is the variation of the horizontal velocity $u$ in the normal direction. This is commonly denoted as $\dot{\gamma}$: $$\dot{\gamma}(x,z) = \dfrac{\partial u}{\partial z}(x,z).$$ Its value at the plate is denoted as $$\dot{\gamma}_0(x) = \dot{\gamma}(x,0).$$ This quantity only depends on the distance $x$ from the edge of the plate, and it is this dependence that we want to investigate. In the figure an angle $\phi(x)$ is indicated. It is related to $\dot{\gamma}(x)$ via $\dot{\gamma}_0(x) = \tan\phi(x)$. Far from the plate, where $z$ is large, the flow is little influenced by the presence of the plate, and so there we may take $(u,w)= (U,0)$. In addition to the dependence on the distance $x$ from the edge, the shear rate also will depend on the viscosity, the velocity $U$, and the density $\rho$. In the steady state we can generally write $$\dot{\gamma}_0 =\dot{\gamma}_0(x;\eta,U,\rho).$$ In the following steps we want to find this relationship as precisely as possible.
