If $g^{1}(x,y,r,s)$ is a function such that $g^{1}(\cdot) \in \mathbb{R}$ and suppose that $\displaystyle \frac{\partial g^1}{\partial x}+a\frac{\partial g^1}{\partial r}+b\frac{\partial g^1}{\partial s}=0$ holds, where $a,b$ are non-zero constants.
How can we show that $g^{1}$ can be written independently of $x$, ie $g^{1}$ is a function $y,r,s$?
This problem comes from the following passage
If you have the equation $\partial_x g(x,y,r,s) =0$, then evidently you have $g$ independent of $x$, i.e. a function of $(y,r,s)$ only
I think that the passage you marked in yellow intends to point out that the solutions of the equation with non-zero $a\ne0\ne b$ are parametrized by three variables (they are not necessarily the same as $(y,r,s)$).
Recall the transport equation in 1D: $$\partial_t y + a\partial_x y =0,\\ y(0,x) = y_0(x), $$ it has the solution $y(t,x) = y_0(x-at)$, i.e. it is parametrized by only one variable.