How does the traffic flow model arrive at the scaled equation?

Since $\rho / \rho_*$ appears in the equation we look for some $u$ of the form $$ u(x,t):= {1 \over \rho _*} \rho (ax,bt) $$ for some constants $a,b$. Now we have $u_t =b\rho _t / \rho _* $. So $\rho _t = \rho _* u_t /b$, and the equation for $u$ becomes $$ \rho _* u_t/b + (V_* \rho _* u(1-u))_x = \rho _t + (V_* \rho (1-\rho /\rho _* ))_x = 0 $$ Now if we choose $b$ such that $\rho _* /b = V_* \rho_*$ then we can take $ V_* \rho_*$ outside the derivative (since it is constant) and factor it out of the equation to get $$  u_t + ( u(1-u))_x=0 $$ as desired. Hence we can set $b=1/V_*$. And since $a$ did not appear in our calculations we set it equal to 1 (i.e. we do not scale w.r.t. $x$). So the final answer becomes $$ u(x,t):= {1 \over \rho _*} \rho (x,t/V_*)$$