I don't understand the 'idea' behind the method of characteristics

Below is an image of my lecture notes explaining the idea behind the method of characteristics for quasilinear first order PDEs.

However I don’t understand how the curve $C_s$ is defined and how it translates into the graph $Gr_u$. How does $C_s$ ‘start’ on the initial curve if there is no $f(s)$ or $g(s)$ in the equation for $C_s$? What does $x(t;s)$ mean, does that mean (say for the $x$ term) that for fixed $s$ one starts at $x(s-\epsilon_s)$ and finishes at $x(s+\epsilon_s)$? If so what is this ‘function’ $x$?

Thanks, I would really appreciate someone explaining this to me! :

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