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If they say that $f_i$ is the central difference discrete of $f(t,x)$ in the point $(t_n,x_i)$.

What do they mean by this?

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Typically a central difference means you compute $f(x_{i+1}) - f(x_{i-1})$ and often you may divide by $\frac{1}{2 \Delta t}$ or something similar. The key thing is that a difference is computed where $f(x_i)$ is in the centre of the two things which we're subtracting.

See http://en.wikipedia.org/wiki/Finite_difference for more information

In your case where you have a function of two variables, you may take $4$ points to average over. There are equivalent notions for higher order functions.

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