You "change variables" in a differential equation using the "chain rule".
That is, $$\frac{\partial u}{\partial x}= \frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x}+ \frac{\partial u}{\partial\tau}\frac{\partial \tau}{\partial x}$$ and $$\frac{\partial u}{\partial t}= \frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial t}+ \frac{\partial u}{\partial\tau}\frac{\partial \tau}{\partial t}$$.
Here, we are given [math]\xi= x- Vt[/math] and [math]\tau= t[/math] so [math]\frac{\partial \xi}{\partial x}= 1[/math] and [math]\frac{\partial \tau}{\partial x}= 0[/math]. [math]\frac{\partial u}{\partial x}= \frac{\partial u}{\partial \xi}[/math].
Similarly [math]\frac{\partial u}{\partial t}= \frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial t}+ \frac{\partial u}{\partial \tau}\frac{\partial \tau}{\partial t}[/math]. Since [math]\frac{\partial \xi}{\partial t}= -V[/math] and [math]\frac{\partial \tau}{\partial t}= 1[/math], [math]\frac{\partial u}{\partial t}= -V\frac{\partial u}{\partial \xi}+ \frac{\partial u}{\partial\tau}[/math].
Repeat that to get the second derivatives.